# A Unified Immersed Finite Element Error Analysis
for One-Dimensional Interface Problems

Slimane Adjerid<EMAIL_ADDRESS>Tao Lin<EMAIL_ADDRESS>Haroun Meghaichi
<EMAIL_ADDRESS>
(Department of Mathematics, Virginia Tech )

###### Abstract

It has been noted that the traditional scaling argument cannot be directly
applied to the error analysis of immersed finite elements (IFE) because, in
general, the spaces on the reference element associated with the IFE spaces on
different interface elements via the standard affine mapping are not the same.
By analyzing a mapping from the involved Sobolev space to the IFE space, this
article is able to extend the scaling argument framework to the error
estimation for the approximation capability of a class of IFE spaces in one
spatial dimension. As demonstrations of the versatility of this unified error
analysis framework, the manuscript applies the proposed scaling argument to
obtain optimal IFE error estimates for a typical first-order linear hyperbolic
interface problem, a second-order elliptic interface problem, and the fourth-
order Euler-Bernoulli beam interface problem, respectively.

keywords: Immersed finite elements, error analysis, Bramble Hilbert lemma,
scaling argument.

##  1 Introduction

Partial differential equations with discontinuous coefficients arise in many
areas of sciences and engineering such as heat transfer, acoustics, structural
mechanics, and electromagnetism. The discontinuity of the coefficients results
in multiple challenges in the design and the analysis of numerical methods and
it is an active area of research in the communities of finite element, finite
volume, as well as finite difference method.

The immersed finite element (IFE) methods can use an interface independent
mesh to solve an interface problem. Many publications were about IFE methods
using either linear [21, 22, 23], bilinear [18, 25], or trilinear polynomials
[17, 33]. IFE methods have been applied to a variety of problems, such as
parabolic interface problems [19, 27, 29], hyperbolic interface problems [6],
the acoustic interface problem [7, 32] and Stokes and Navier-Stokes interface
problems [3, 13, 20, 34]. IFE methods with higher degree polynomials have also
been explored [2, 4, 5, 13, 16, 28]. In particular, Adjerid and Lin [5]
constructed IFE spaces of arbitrary degree and analyzed their approximation
capabilities.

In [7, 32], Adjerid and Moon discussed IFE methods for the following acoustic
interface problem

$\begin{cases}p_{t}(x,t)=\rho(x)c(x)^{2}v_{x}(x,t),&x\in(a,\alpha)\cup(\alpha,b),\\\
\rho(x)v_{t}(x,t)=p_{x}(x,t),&x\in(a,\alpha)\cup(\alpha,b),\\\
[v]_{x=\alpha}=[p]_{x=\alpha}=0.\end{cases}$ (1)

where $\rho,c$ are equal to $\rho_{+},c_{+}$ on interval $(\alpha,b)$ and to
$\rho_{-},c_{-}$ on $(a,\alpha)$. Assuming that the exact solution $(u,p)$ has
sufficient regularity in $(a,\alpha)$ and $(\alpha,b)$, respectively, we can
follow the idea in [31] to show that the exact solution satisfies the
following so-called extended jump conditions:

$\frac{\partial^{k}}{\partial
x^{k}}p(\alpha^{+},t)=r^{p}_{k}\frac{\partial^{k}}{\partial
x^{k}}p(\alpha^{-},t),\qquad\frac{\partial^{k}}{\partial
x^{k}}v(\alpha^{+},t)=r^{v}_{k}\frac{\partial^{k}}{\partial
x^{k}}v(\alpha^{-},t),\qquad k=0,1,\dots,$ (2)

for certain positive constants $r^{p}_{k}$ and $r^{v}_{k}$. In [7, 32], IFE
spaces based on polynomials of degree up to $4$ were developed with these
extended jump conditions, and these IFE spaces were used with a discontinuous
Galerkin (DG) method to solve the above acoustic interface problem with
pertinent initial and boundary conditions. Numerical examples presented in [7,
32] demonstrated the optimal convergence of this DG IFE method, but we have
not seen any error analysis about it in the related literature.

Extended jump conditions have also been used in the development of higher
degree IFE spaces for solving other interface problems [2, 4, 5, 13, 16, 28].
This motivates us to look for a unified framework for the error analysis for
methods based on IFE spaces constructed with the extended jump conditions such
as those in (2). As an initial effort, our focus here is on one-dimensional
interface problems.

One challenge in error estimation for IFE methods is that the scaling argument
commonly used in error estimation for traditional finite element methods
cannot be directly applied. In the standard scaling argument, local finite
element spaces on elements in a sequence of meshes are mapped to the same
finite element space on the reference element via an affine transformation.
However, the same affine transformation will map the local IFE spaces on
interface elements in a sequence of meshes to different IFE spaces on the
reference element because of the variation of interface location in the
reference element, see the illustration in Figure 1. A straightforward
application of the scaling argument to the analysis of the approximation
capability of an IFE space will result in error bounds of a form
$C(\check{\alpha})h^{r}$, i.e., the constant factor $C(\check{\alpha})$ in the
derived error bounds depend on the location of the interface in the reference
element, and this kind of error bounds cannot be used to show the convergence
of the related IFE method unless one can show that the constant factor
$C(\check{\alpha})$ is bounded for all $\check{\alpha}$ in the reference
element, which, to our best knowledge, is difficult to establish. Alternative
analysis techniques such as multi-point Taylor expansions are used [5] which
becomes awkward for higher degree IFE spaces, particularly so for higher
degree IFE spaces in higher dimension. To circumvent this predicament of the
classical scaling argument, we introduce a mapping between the related Sobolev
space and the IFE space by using weighted averages of the derivatives in terms
of the coefficients in the jump conditions. We show that the Sobolev norm of
the error of this mapping can be bounded by the related Sobolev semi-norm.
This essential property enables us to establish a Bramble-Hilbert type lemma
for the IFE spaces, and, to our best knowledge, this is the first result that
makes the scaling argument applicable in the error analysis of a class of IFE
methods. For demonstrating the versatility of this unified error analysis
framework, we apply it to establish, for the first time, the optimal
approximation capability of the IFE space designed for the acoustic interface
problem (1). Similarly, we apply this immersed scaling argument to the IFE
space designed for an elliptic interface problem considered in [5] as well as
the IFE space for the Euler-Bernoulli Beam interface problem considered in
[24, 26, 35] leading to much simpler and elegant proofs.

Figure 1: The relative position of the interface (on the right) changes as the
we refine the mesh (on the left).

The paper is organized as follows. In Section 2, we introduce the notation and
spaces used in the rest of the paper. In Section 3, we restrict ourselves to
study of the IFE functions on the interval $[0,1]$, we show that they have
similar properties to polynomials, for example, they both have the same
maximum number of roots, they both admit a Lagrange basis and they both
satisfy an inverse and a trace inequality. In Section 4, we define the notion
of uniformly bounded RIFE operators and how the scaling argument is applicable
using an immersed Bramble-Hilbert lemma. In Section 5, we study the
convergence of the DG-IFE method for the acoustic interface problem (1). In
Section 6, we give shorter and simpler proofs for the optimal convergence of
IFE methods for the second-order elliptic interface problem as well as the
fourth-order Euler-Bernoulli beam interface problem.

##  2 Preliminaries

Throughout the article, we will consider a bounded open interval $I=(a,b)$
with $\left|a\right|,\left|b\right|<\infty$, and let $\alpha\in I$ be the
interface point dividing $I$ into two open intervals
$I^{-}=(a,\alpha),I^{+}=(\alpha,b)$. This convention extends to any other open
interval $B\subseteq\mathbb{R}$ with $B^{-}=B\cap(-\infty,\alpha)$ and
$B^{+}=B\cap(\alpha,\infty)$. For every bounded open interval $B$ not
containing $\alpha$, let $W^{m,p}(B)$ be the usual Sobolev space on $B$
equipped with the norm $\left\lVert\cdot\right\rVert_{m,p,B}$ and the seminorm
$|\cdot|_{m,p,B}$. We are particularly interested in the case of $p=2$
corresponding to the Hilbert space $H^{m}(B)=W^{m,2}(B)$, and we will use
$\left\lVert\cdot\right\rVert_{m,B}$ and $|\cdot|_{m,B}$ to denote
$\left\lVert\cdot\right\rVert_{m,2,B},|\cdot|_{m,2,B}$ respectively for
convenience. We will use $(\cdot,\cdot)_{B}$ and $(\cdot,\cdot)_{w,B}$ to
denote the classical and the weighted $L^{2}$ inner product defined as

$(f,g)_{B}=\int_{B}f(x)g(x)\ dx,\qquad(f,g)_{w,B}=\int_{B}w(x)f(x)g(x)\
dx,\qquad w(x)>0,\ \forall\ x\in B.$

Given a positive finite sequence
$r=(r_{i})_{i=0}^{\tilde{m}},~{}{\tilde{m}}\geq 0$ and an open interval $B$
containing $\alpha$, we introduce the following piecewise Sobolev space:

$\displaystyle\mathcal{H}^{m+1}_{\alpha,r}(B)$ $\displaystyle=\left\\{u\mid
u_{\mid B^{\pm}}\in
H^{m{+1}}(B^{\pm}),u^{(k)}(\alpha^{+})=r_{k}u^{(k)}(\alpha^{-}),\ \forall
k=0,1,\dots,m\right\\},~{}0\leq m\leq{\tilde{m}}.$ (3)

The norms, semi-norms and the inner product that we will use on
$\mathcal{H}^{m+1}_{\alpha,r}(B)$ are

$\left\lVert\cdot\right\rVert_{m+1,B}=\sqrt{\left\lVert\cdot\right\rVert_{m+1,B^{-}}^{2}+\left\lVert\cdot\right\rVert_{m+1,B^{+}}^{2}},~{}~{}\left|\cdot\right|_{m,B}=\sqrt{\left|\cdot\right|_{m,B^{-}}^{2}+\left|\cdot\right|_{m,B^{+}}^{2}},~{}~{}\left(f,g\right)_{w,B}=\left(f,g\right)_{w,B^{-}}+\left(f,g\right)_{w,B^{+}}.$

We note, by the Sobolev embedding theory, that
$\mathcal{H}^{m+1}_{\alpha,r}(B)$ is a subspace of

$\displaystyle\mathcal{C}^{m}_{\alpha,r}(\overline{B})$
$\displaystyle=\left\\{u\mid u_{\mid B^{\pm}}\in
C^{m}(\overline{B^{\pm}}),u^{(k)}(\alpha^{+})=r_{k}u^{(k)}(\alpha^{-}),\
\forall k=0,1,\dots,m\right\\},~{}0\leq m\leq{\tilde{m}}.\ $ (4)

By dividing $I$ into $N$ sub-intervals, we obtain the following partition of
$I$:

$I_{k}=(x_{k-1},x_{k}),\quad\mathcal{T}_{h}=\\{I_{k}\\}_{k=1}^{N},\quad
a=x_{0}<x_{1}<\dots<x_{N}=b,\qquad h=\max_{1\leq k\leq N}(x_{k}-x_{k-1}).$

We will assume that there is $k_{0}\in\\{1,2,\dots,N\\}$ such that
$x_{k_{0}-1}<\alpha<x_{k_{0}}$, which is equivalent to
$\alpha\in\overset{\circ}{I}_{k_{0}}$. We define the discontinuous immersed
finite element space $W^{m}_{\alpha,r}$ on the interval $I$ as

${\cal
W}^{m}_{\alpha,r}(\mathcal{T}_{h})=\left\\{\varphi\mid\varphi_{\mid{I_{k}}}\in\mathcal{P}^{m}(I_{k})\text{
for }k\in\\{1,\dots,N\\}\backslash\\{k_{0}\\}\text{ and }\varphi_{\mid
I_{k_{0}}}\in\mathcal{V}^{m}_{\alpha,r}(I_{k_{0}})\right\\},$ (5)

where $\mathcal{P}^{m}(I_{k})$ is the space of polynomials of degree at most
$m$ on $I_{k}$ and $\mathcal{V}^{m}_{\alpha,r}(I_{k_{0}})$ is the local
immersed finite element (LIFE) space defined as:

$\displaystyle\mathcal{V}^{m}_{\alpha,r}(I_{k_{0}})=\left\\{\varphi\in\mathcal{C}^{m}_{\alpha,r}(I_{k_{0}})\mid\varphi_{\mid
I_{k_{0}}^{s}}\in\mathcal{P}^{m}\left(I_{k_{0}}^{s}\right),\
s=+,-\right\\},~{}0\leq m\leq{\tilde{m}}.$ (6)

In discussions from now on, given a function $v$ in
$\mathcal{H}^{m+1}_{\alpha,r}$ (or $\mathcal{C}^{m}_{\alpha,r}$ or
$\mathcal{V}^{m}_{\alpha,r}$), its derivative is understood in the piecewise
sense unless specified otherwise. By definition, we can readily verify that
$\mathcal{V}^{m-1}_{\alpha,r}(I_{k_{0}})\subset\mathcal{V}^{m}_{\alpha,r}(I_{k_{0}})$
for a given finite sequence $r=(r_{i})_{i=0}^{\tilde{m}},~{}{\tilde{m}}\geq
1$. In order to study the LIFE space $\mathcal{V}^{m}_{\alpha,r}(I_{k_{0}})$,
we will investigate the properties of the corresponding reference IFE (RIFE)
space $\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$ on the reference
interval $\check{I}=[0,1]$ with an interface point $\check{\alpha}\in(0,1)$.
Our goal is to extend the scaling argument to such IFE spaces and use the IFE
scaling argument to show IFE spaces such as
$\mathcal{V}^{m}_{\alpha,r}(I_{k_{0}})$ have the optimal approximation
capability, i.e., every function in $\mathcal{H}^{m+1}_{\alpha,r}(I_{k_{0}})$
can be approximated by functions from the IFE space
$\mathcal{V}^{m}_{\alpha,r}(I_{k_{0}})$ at the optimal convergence rate.

Following the convention in the error analysis literature for finite element
methods, we will often use a generic constant $C$ in estimates whose value
varies depending on the context, but this generic constant is independent of
$h$ and the interface $\alpha\in I_{k_{0}}$ or $\check{\alpha}\in\check{I}$
unless otherwise declared.

##  3 Properties of the RIFE space
$\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$

For a given function
$\check{\varphi}\in\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$, we will
write $\check{\varphi}=(\check{\varphi}_{-},\check{\varphi}_{+})$ where
$\check{\varphi}_{s}=\check{\varphi}_{\mid\check{I}^{s}}\in\mathcal{P}^{m}(\check{I}^{s})$
for $s=+,-$. Additionally, we will use
$\check{\varphi}_{s}^{(k)}(\check{\alpha})$ to denote
$\lim_{x\to\check{\alpha}^{s}}\check{\varphi}_{s}^{(k)}(x),~{}s=\pm$ for a
given integer $k\geq 0$. For clarity, we will use $s^{\prime}$ to denote the
dual of $s$, i.e., if $s=\pm$, then $s^{\prime}=\mp$.

######  Lemma 1.

Let ${\tilde{m}}\geq m\geq
0,\\{r_{k}\\}_{k=0}^{\tilde{m}}\subset\mathbb{R}_{+},\check{\alpha}\in(0,1)$
and $s\in\\{+,-\\}$. The following statements hold

  1. 1.

For every $\check{\varphi}_{s}\in\mathcal{P}^{m}(\check{I}^{s})$ there is a
unique
$\check{\varphi}_{s^{\prime}}\in\mathcal{P}^{m}(\check{I}^{s^{\prime}})$ such
that
$\check{\varphi}=(\check{\varphi}_{-},\check{\varphi}_{+})\in\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$.

  2. 2.

The dimension of $\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$ is $m+1$.

  3. 3.

The set $\\{\mathcal{N}^{k}_{\check{\alpha},r}\\}_{k=0}^{m}$, where

$\mathcal{N}^{k}_{\check{\alpha},r}(x)=\begin{cases}(x-\check{\alpha})^{k},&x\in\check{I}^{-},\\\
r_{k}(x-\check{\alpha})^{k},&x\in\check{I}^{+},\end{cases}$ (7)

forms a basis of $\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$ and will be
referred to as the canonical basis.

###### Proof.

We will prove the statements in order:

  1. 1.

Let $\check{\varphi}_{\pm}\in\mathcal{P}^{m}(\check{I}^{\pm})$, then
$(\check{\varphi}_{-},\check{\varphi}_{+})\in\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$
if and only if

$\check{\varphi}_{\mp}^{(k)}(\check{\alpha})=\left(r_{k}\right)^{\mp
1}\check{\varphi}_{\pm}^{(k)}(\check{\alpha}),\qquad k=0,1,\dots,m,$

which uniquely defines a polynomial
$\check{\varphi}_{\mp}\in\mathcal{P}^{m}(\check{I}^{\mp})$:

$\check{\varphi}_{\mp}(x)=\sum_{k=0}^{m}\frac{\left(r_{k}\right)^{\mp
1}\check{\varphi}_{\pm}^{(k)}(\check{\alpha})}{k!}(x-\check{\alpha})^{k}.$

  2. 2.

We have shown that the maps $\check{\varphi}_{-}\mapsto\check{\varphi}$ is
well defined and injective which implies that the map
$\check{\varphi}\mapsto\check{\varphi}_{-}$ is surjective since every
$\check{\varphi}_{-}\in\mathcal{P}^{m}(\check{I}^{-})$ can be extended to
$\check{\varphi}\in\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$. Hence,
$\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$ is isomorphic to
$\mathcal{P}^{m}(\check{I}^{-})$ implying that the dimension of
$\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$ is $m+1$.

  3. 3.

We only need to show that $\\{\mathcal{N}^{k}_{\check{\alpha},r}\\}_{k=0}^{m}$
is linearly independent: Assume that
$\check{\varphi}=\sum_{k=0}^{m}c_{k}\mathcal{N}^{k}_{\check{\alpha},r}\equiv
0$, then $\check{\varphi}_{-}\equiv 0$ which implies that $c_{k}=0$ for all
$k=0,1,\dots,m$.

∎

The results in Lemma 1 allows us to introduce an extension operator that maps
$\check{\varphi}_{s}$ to $\check{\varphi}_{s^{\prime}}$.

######  Definition 1.

Let ${\tilde{m}}\geq m\geq
0,\\{r_{k}\\}_{k=0}^{\tilde{m}}\subset\mathbb{R}_{+},\check{\alpha}\in(0,1)$
and $s\in\\{+,-\\}$. We define the extension operator
$\mathcal{E}^{m,s^{\prime}}_{\check{\alpha},r}:\mathcal{P}^{m}(\check{I}^{s})\to\mathcal{P}^{m}(\check{I}^{s^{\prime}})$
that maps every $\check{\varphi}_{s}\in\mathcal{P}^{m}(\check{I}^{s})$ to
$\mathcal{E}^{m,s^{\prime}}_{\check{\alpha},r}(\check{\varphi}_{s})=\check{\varphi}_{s^{\prime}}$
such that
$(\check{\varphi}_{-},\check{\varphi}_{+})\in\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$.

By Lemma 1, the extension operator
$\mathcal{E}^{m,s^{\prime}}_{\check{\alpha},r}$ is well-defined and is linear.
Furthermore, by Lemma 1 again, this extension operator is also invertible.
Consequently, the dimension of the RIFE space is the same as the dimension of
the traditional polynomial space of the same degree.

Next, we will estimate the operator norm of
$\mathcal{E}^{m,s^{\prime}}_{\check{\alpha},r}$. Let
$\check{h}_{-}=\check{\alpha},\ \check{h}_{+}=1-\check{\alpha}$ be the lengths
of the sub-intervals $\check{I}^{\pm}$ formed by $\check{\alpha}$. First, let
us consider the following example
$\check{\varphi}=(\check{\varphi}_{-},\check{\varphi}_{+})=\mathcal{N}^{m}_{\check{\alpha},r}$
defined by (7), we have

$\left\lVert\mathcal{E}^{m,s^{\prime}}_{\check{\alpha},r}(\check{\varphi}_{-})\right\rVert_{0,\check{I}^{+}}=\left\lVert\check{\varphi}_{+}\right\rVert_{0,\check{I}^{+}}=|r_{m}|\left(\frac{\check{h}_{+}}{\check{h}_{-}}\right)^{m}\sqrt{\frac{\check{h}_{+}}{\check{h}_{-}}}\left\lVert\check{\varphi}_{-}\right\rVert_{0,\check{I}^{-}},\qquad\text{where
}\check{h}_{-}=\check{\alpha},\ \check{h}_{+}=1-\check{\alpha}.$

Hence, if $h_{-}>h_{+}$, we get
$\left\lVert\check{\varphi}_{+}\right\rVert_{0,\check{I}^{+}}\leq|r_{m}|\left\lVert\check{\varphi}_{-}\right\rVert_{0,\check{I}^{-}}$.
In the following lemma, we will show that a similar result holds for all
$\check{\varphi}\in\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$.
Consequently, for every interface position $\check{\alpha}\in I$, one of the
two extension operators
$\mathcal{E}^{m,s^{\prime}}_{\check{\alpha},r},~{}s^{\prime}=-,+$ will be
bounded independently of $\check{\alpha}$.

######  Lemma 2.

There exists a constant $C>0$ that depends on $m$ such that for every
$(\check{\varphi}_{-},\check{\varphi}_{+})\in\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$,
we have

$\left\lVert\mathcal{E}^{m,s^{\prime}}_{\check{\alpha},r}\check{\varphi}_{s}\right\rVert_{0,\check{I}^{s^{\prime}}}=\left\lVert\check{\varphi}_{s^{\prime}}\right\rVert_{0,\check{I}^{s^{\prime}}}\leq
C\sqrt{\frac{\check{h}_{s^{\prime}}}{\check{h}_{s}}}\left(\max_{0\leq i\leq
m}r_{i}^{{\tilde{s}}}\right)\max\left(1,\left(\frac{\check{h}_{s^{\prime}}}{\check{h}_{s}}\right)^{m}\right)\left\lVert\check{\varphi}_{s}\right\rVert_{0,\check{I}^{s}},\qquad
s=+,-,$ (8)

where
${\tilde{s}}={\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\mp}1$
for $s=\pm$. In particular, if $\check{h}_{s}\geq\check{h}_{s^{\prime}}$, we
have

$\left\lVert\check{\varphi}_{s^{\prime}}\right\rVert_{0,\check{I}^{s^{\prime}}}\leq
C\sqrt{\check{h}_{s^{\prime}}}\left(\max_{0\leq i\leq
m}r_{i}^{{\tilde{s}}}\right)\left\lVert\check{\varphi}_{s}\right\rVert_{0,\check{I}^{s}}.$
(9)

###### Proof.

First, we note that (9) is a straightforward consequence of (8). Here, we only
need prove (8) for $s=-$ since the case $s=+$ can be proven similarly. For
every
$(\check{\varphi}_{-},\check{\varphi}_{+})\in\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$,
we first define $\hat{\varphi}_{-}\in\mathcal{P}^{m}([0,1])$ as
$\hat{\varphi}_{-}(\xi)=\check{\varphi}_{-}(\check{h}_{-}\xi)$ which yields

$\hat{\varphi}_{-}^{(i)}(1)=\check{h}_{-}^{i}\check{\varphi}_{-}^{(i)}(\check{\alpha}),\quad
i=0,1,\dots,m.$ (10)

Now, let us write $\check{\varphi}_{+}$ as a finite Taylor sum around
$\check{\alpha}$ and use
$\check{\varphi}_{+}^{(i)}(\check{\alpha})=r_{i}\check{\varphi}_{-}^{(i)}(\check{\alpha})$
to obtain:

$\check{\varphi}_{+}(x)=\sum_{i=0}^{m}\check{\varphi}_{+}^{(i)}(\check{\alpha})\frac{(x-\check{\alpha})^{i}}{i!}=\sum_{i=0}^{m}r_{i}\check{\varphi}_{-}^{(i)}(\check{\alpha})\frac{(x-\check{\alpha})^{i}}{i!}.$

Using (10), we can replace $\check{\varphi}_{-}^{(i)}(\check{\alpha})$ by
$\check{h}_{-}^{-i}\hat{\varphi}_{-}^{(i)}(1)$:

$\check{\varphi}_{+}(x)=\sum_{i=0}^{m}r_{i}\hat{\varphi}_{-}^{(i)}(1)\check{h}_{-}^{-i}\frac{(x-\check{\alpha})^{i}}{i!}.$
(11)

We square and integrate (11), then we apply the change of variables
$z=x-\check{\alpha}$ to get

$\left\lVert\check{\varphi}_{+}\right\rVert_{0,\check{I}^{+}}^{2}=\int_{\check{\alpha}}^{1}\left(\sum_{i=0}^{m}r_{i}\hat{\varphi}_{-}^{(i)}(1)\check{h}_{-}^{-i}\frac{(x-\check{\alpha})^{i}}{i!}\right)^{2}\
dx=\int_{0}^{\check{h}_{+}}\left(\sum_{i=0}^{m}r_{i}\hat{\varphi}_{-}^{(i)}(1)\left(\frac{z}{\check{h}_{-}}\right)^{i}\frac{1}{i!}\right)^{2}\
dz.$

We can bound $r_{i}$ and $|\hat{\varphi}_{-}^{(i)}(1)|$ by their maximum
values for $0\leq i\leq m$ and we can bound
$\left(\frac{z}{\check{h}_{-}}\right)^{i}$ by

$\left(\frac{z}{\check{h}_{-}}\right)^{i}\leq\max\left(1,\left(\frac{\check{h}_{+}}{\check{h}_{-}}\right)^{m}\right).$

We also have $\sum_{i=0}^{m}\frac{1}{i!}\leq e$. Using these bounds, we get

$\displaystyle\left\lVert\check{\varphi}_{+}\right\rVert_{0,\check{I}^{+}}^{2}$
$\displaystyle\leq\left(\max_{0\leq i\leq m}r_{i}\right)^{2}\left(\max_{0\leq
i\leq
m}|\hat{\varphi}_{-}^{(i)}(1)|\right)^{2}\max\left(1,\left(\frac{\check{h}_{+}}{\check{h}_{-}}\right)^{m}\right)^{2}\int_{0}^{\check{h}_{+}}e^{2}\
dz$ $\displaystyle=\left(\max_{0\leq i\leq m}r_{i}\right)^{2}\left(\max_{0\leq
i\leq
m}|\hat{\varphi}_{-}^{(i)}(1)|\right)^{2}\max\left(1,\left(\frac{\check{h}_{+}}{\check{h}_{-}}\right)^{m}\right)^{2}\check{h}_{+}e^{2}.$
(12)

Since $\mathcal{P}^{m}([0,1])$ is a finite dimensional space, all norms are
equivalent. In particular, there is a constant $C(m)$ such that

$\left(\max_{0\leq i\leq m}|p^{(i)}(1)|\right)\leq C(m)\left\lVert
p\right\rVert_{0,[0,1]},~{}\forall p\in\mathcal{P}^{m}([0,1]),$

which leads to

$\left(\max_{0\leq i\leq m}|\hat{\varphi}_{-}^{(i)}(1)|\right)^{2}\leq
C(m)\left\lVert\hat{\varphi}_{-}\right\rVert_{0,[0,1]}^{2}.$ (13)

By using a change of variables, we can show that

$\left\lVert\hat{\varphi}_{-}\right\rVert_{0,[0,1]}^{2}=\frac{1}{\check{h}_{-}}\left\lVert\check{\varphi}_{-}\right\rVert_{0,\check{I}^{-}}^{2}.$
(14)

Finally, we combine (12), (13) and (14) to get

$\left\lVert\mathcal{E}_{\check{\alpha},r}^{m,+}\check{\varphi}_{-}\right\rVert^{2}=\left\lVert\check{\varphi}_{+}\right\rVert_{0,\check{I}^{+}}^{2}\leq
C(m)\left(\max_{0\leq i\leq
m}r_{i}\right)^{2}\left(\frac{\check{h}_{+}}{\check{h}_{-}}\right)\max\left(1,\left(\frac{\check{h}_{+}}{\check{h}_{-}}\right)^{m}\right)^{2}\left\lVert\check{\varphi}_{-}\right\rVert_{0,\check{I}^{-}}^{2}$

which is (8) for $s=-$.

∎

Next, we will use the bounds on the extension operator
$\mathcal{E}^{m,s}_{\check{\alpha},r}$ to establish inverse inequalities which
are independent of $\check{\alpha}$ for the RIFE space.

######  Lemma 3.

Let $\tilde{m}\geq m\geq
0,\\{r_{k}\\}_{k=0}^{\tilde{m}}\subset\mathbb{R}_{+}$,
$\check{\alpha}\in(0,1)$. Then there exists $C(m,r)>0$ independent of
$\check{\alpha}$ such that for every
$\check{\varphi}\in\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$ we have

$\displaystyle|\check{\varphi}|_{i,\check{I}}\leq
C(m,r)\left\lVert\check{\varphi}\right\rVert_{0,\check{I}},~{}0\leq i\leq
m+1.$ (15)

###### Proof.

The estimate given in (15) obviously holds for $i=0$ and $i=m+1$. Without loss
of generality, assume that $\check{h}_{-}\geq\check{h}_{+}$, this implies that
$\check{h}_{-}\geq\frac{1}{2}$. Then, using the classical inverse inequality
[11] we have

$\left\lVert\check{\varphi}^{\prime}_{-}\right\rVert_{0,\check{I}^{-}}\leq
C\check{h}_{-}^{-1}\left\lVert\check{\varphi}_{-}\right\rVert_{0,\check{I}^{-}}\leq
2C\left\lVert\check{\varphi}_{-}\right\rVert_{0,\check{I}^{-}}.$ (16)

By the Taylor expansion of $\check{\varphi}^{\prime}_{+}(x)$ at
$x=\check{\alpha}$, we have

$\check{\varphi}^{\prime}_{+}=\mathcal{E}^{m-1,+}_{\check{\alpha},\tau(r)}\left(\frac{\mathrm{d}}{\mathrm{d}x}\check{\varphi}_{-}\right),$
(17)

where $\tau:(r_{0},r_{1},\dots,r_{m})\mapsto(r_{1},\dots,r_{m})$ is the shift
operator. By (9) and the inverse inequality given in (16), we have

$\left\lVert\check{\varphi}^{\prime}_{+}\right\rVert_{0,\check{I}^{+}}\leq
C(m)\left(\max_{1\leq i\leq
m}r_{i}\right)\left\lVert\check{\varphi}_{-}^{\prime}\right\rVert_{0,\check{I}^{-}}\leq
C(m,r)\left\lVert\check{\varphi}_{-}\right\rVert_{0,\check{I}^{-}}.$ (18)

Therefore, we have

$\left|\check{\varphi}\right|_{1,\check{I}}=\left\lVert\check{\varphi}^{\prime}\right\rVert_{0,\check{I}}\leq
C(m,r)\left\lVert\check{\varphi}_{-}\right\rVert_{0,\check{I}^{-}}\leq
C(m,r)\left\lVert\check{\varphi}\right\rVert_{0,\check{I}}$

which proves (15) for $i=1$. Applying similar arguments, we can prove (15) for
other values of $i$.

∎

Since
$\check{\varphi}_{+}=\mathcal{E}^{m,+}_{\check{\alpha},r}\left(\check{\varphi}_{-}\right)$,
the formula in (17) leads to the following identity about the permutation of
the classical differential operator and the extension operator:

$\frac{\mathrm{d}}{\mathrm{d}x}\mathcal{E}^{m,+}_{\check{\alpha},r}\left(\check{\varphi}_{-}\right)=\check{\varphi}^{\prime}_{+}=\mathcal{E}^{m-1,+}_{\check{\alpha},\tau(r)}\left(\frac{\mathrm{d}}{\mathrm{d}x}\check{\varphi}_{-}\right),~{}\forall
m\geq 1.$ (19)

As a piecewise function
$\check{\varphi}=(\check{\varphi}_{-},\check{\varphi}_{+})\in\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$,
the value of $\check{\varphi}$ at $\check{\alpha}$ is not defined in general
since the two sided limits $\check{\varphi}_{-}(\check{\alpha})$ and
$\check{\varphi}_{+}(\check{\alpha})$ could be different if $r_{0}\neq 1$.
However, if $\check{\varphi}_{s}(\check{\alpha})=0$ then
$\check{\varphi}_{s^{\prime}}(\check{\alpha})=0$ for $s=+,-$. Furthermore, the
multiplicity of $\check{\alpha}$ as a root of $\check{\varphi}_{-}$ is the
same as its multiplicity as a root of $\check{\varphi}_{+}$. This observation
motivates to define $\check{\alpha}$ as a root of $\check{\varphi}$ of
multiplicity $d$ if $\check{\alpha}$ is a root of $\check{\varphi}_{-}$ of
multiplicity $d$. The following theorem shows that the number of roots of a
non-zero function
$\check{\varphi}\in\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$ counting
multiplicities cannot exceed $m$ (similar to a polynomial of degree $m$), this
theorem will be crucial to establish the existence of a Lagrange-type basis in
$\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$ and constructing an immersed
Radau projection later in Section 5.

In the discussions below, we will omit the phrase “counting multiplicities”
for the sake of conciseness. For example, we say that $(x-2)^{2}$ has two
roots in $\mathbb{R}$.

######  Theorem 1.

For ${\tilde{m}}\geq m\geq
0,\\{r_{k}\\}_{k=0}^{\tilde{m}}\subset\mathbb{R}_{+}$, and
$\check{\alpha}\in(0,1)$, every non-zero
$\check{\varphi}\in\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$ has at most
$m$ roots.

###### Proof.

We start from the base case $m=0$ and then proceed by induction. Let
$\check{\varphi}\in\mathcal{V}^{0}_{\check{\alpha},r}(\check{I})$. if
$\check{\varphi}\not\equiv 0$ then
$\check{\varphi}=c\mathcal{N}^{0}_{\check{\alpha},r}$ for some $c\neq 0$. In
this case, $\check{\varphi}$ has no roots since
$\mathcal{N}^{0}_{\check{\alpha},r}$ has no roots. Now, assume that for every
positive sequence $q$, the number of roots of any non-zero
$\check{\varphi}\in\mathcal{V}^{m-1}_{\check{\alpha},q}(\check{I})$ is at most
$m-1$. Next, we will show that for a given positive sequence $r$, every
function $\check{\varphi}\in\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$ has
at most $m$ roots by contradiction:

Assume that
$\check{\varphi}=(\check{\varphi}_{-},\check{\varphi}_{+})\in\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$
is a non-zero function that has $j$ disctinct roots $\\{\xi_{i}\\}_{i=1}^{j}$
of multiplicities $\\{d_{i}\\}_{i=1}^{j}$ such that
$D=d_{1}+d_{2}+\dots+d_{j}>m$. Therefore, $\xi_{j}>\check{\alpha}$ and
$\xi_{1}\leq\check{\alpha}$ because
$\check{\varphi}_{\pm}\in\mathcal{P}^{m}(\check{I}^{s})$. let $\xi_{i_{0}}$ be
the largest root that is not larger than $\check{\alpha}$, i.e.,

$0\leq\xi_{1}<\xi_{2}<\dots<\xi_{i_{0}}\leq\check{\alpha}<\xi_{i_{0}+1}<\dots<\xi_{j}\leq
1,$

By the definition of
$\check{\varphi}=(\check{\varphi}_{-},\check{\varphi}_{+})$,
$\check{\varphi}_{-}$ has $D_{1}=d_{1}+d_{2}+\dots+d_{i_{0}}$ roots in
$[\xi_{1},\xi_{i_{0}}]$ and $\check{\varphi}_{+}$ has
$D_{2}=d_{i_{0}+1}+d_{i_{0}+2}+\dots+d_{j}$ roots in
$[\xi_{i_{0}+1},\xi_{j}]$. Therefore, $\check{\varphi}_{-}^{\prime}$ has
$D_{1}-1$ roots in $[\xi_{1},\xi_{i_{0}}]$ and $\check{\varphi}_{+}^{\prime}$
has $D_{2}-1$ roots in $[\xi_{i_{0}+1},\xi_{j}]$. It remains to show that
$\check{\varphi}^{\prime}$ has an additional root in
$(\xi_{i_{0}},\xi_{i_{0}+1})$. To show that,we consider two cases:

  * •

$\xi_{i_{0}}=\check{\alpha}$: In this case, $\check{\varphi}$ is continuous
and
$\check{\varphi}_{+}(\check{\alpha})=\check{\varphi}_{+}(\xi_{i_{0}+1})=0$. By
the mean value theorem, we conclude that $\check{\varphi}_{+}^{\prime}$ has a
root in $(\xi_{i_{0}},\xi_{i_{0}+1})$.

  * •

$\xi_{i_{0}}<\check{\alpha}$: Assume that $\check{\varphi}^{\prime}(x)>0$ for
all $x\in(\xi_{i_{0}},\xi_{i_{0}+1})\backslash\\{\check{\alpha}\\}$, then
$\check{\varphi}_{-}(\check{\alpha})>0$. Since $r_{0}>0$, we have
$\check{\varphi}_{+}(\check{\alpha})>0$. By integrating
$\check{\varphi}_{+}^{\prime}$ from $\check{\alpha}$ to $\xi_{i_{0}+1}$, we
get $0=\check{\varphi}_{+}(\xi_{i_{0}+1})>0$, a contradiction. A similar
conclusion follows if we assume that $\check{\varphi}^{\prime}(x)<0$ for all
$x\in(\xi_{i_{0}},\xi_{i_{0}+1})\backslash\\{\check{\alpha}\\}$. Therefore,
$\check{\varphi}^{\prime}$ changes sign at some
$x_{0}\in(\xi_{i_{0}},\xi_{i_{0}+1})$. Since, $\check{\varphi}^{\prime}$ does
not change sign at $\check{\alpha}$, then $x_{0}\neq\check{\alpha}$ and
$\check{\varphi}^{\prime}(x_{0})=0$ (because $\check{\varphi}^{\prime}$ is
continuous at $x_{0}$).

In either case above, $\check{\varphi}^{\prime}$ has
$(D_{1}-1)+(D_{2}-1)+1=D-1>m-1$ roots in $\check{I}$ which contradicts the
induction hypothesis since
$\check{\varphi}^{\prime}\in\mathcal{V}^{m-1}_{\check{\alpha},\tau(r)}(\check{I})$.
Therefore, $\check{\varphi}$ has at most $m$ roots.

∎

The previous theorem allows us to establish the existence of a Lagrange basis
on $\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$ for every choice of nodes
and for every degree $m$ which was proved by Moon in [32] for a few specific
cases $m=1,2,3,4$.

######  Theorem 2.

Let ${\tilde{m}}\geq m\geq
0,\\{r_{k}\\}_{k=0}^{\tilde{m}}\subset\mathbb{R}_{+}$, and
$\check{\alpha}\in(0,1)$. Assume $\xi_{0},\xi_{1},\dots,\xi_{m}$ are $m+1$
distinct points in $\check{I}$, then there is a Lagrange basis
$\\{L_{i}\\}_{i=0}^{m}$ of $\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$
that satisfies

$\displaystyle L_{i}(\xi_{j})=\delta_{i,j},\qquad\ 0\leq i,j\leq m.$ (20)

###### Proof.

For each $0\leq i\leq m$, we construct
$\tilde{L}_{i}\in\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$ such that
$\tilde{L}_{i}(\xi_{j})=0$ for all $j\neq i$ by writing $\tilde{L}_{i}$ as

$\tilde{L}_{i}=\sum_{i=0}^{m}a_{i}\mathcal{N}^{i}_{\check{\alpha},r},$

for some $\\{a_{i}\\}_{i=0}^{m}$ chosen such that

$\tilde{L}_{i}(\xi_{j})=\sum_{i=0}^{m}a_{i}\mathcal{N}^{i}_{\check{\alpha},r}(\xi_{j})=0,\quad\forall
j\in\\{0,1,\dots,i-1,i+1,\dots,m\\}.$ (21)

The equations (21) form a homogeneous system of $m$ equations with $m+1$
unknowns. Therefore, it has a non-zero solution. From Theorem 1, we know that
$\tilde{L}_{i}(\xi_{i})\neq 0$; otherwise, $\tilde{L}_{i}$ would have $m+1$
roots. This allows us to define $L_{i}$ as

$L_{i}(x)=\frac{1}{\tilde{L}_{i}(\xi_{i})}\tilde{L}(x).$

By (20), $L_{i},0\leq i\leq m$ are linearly independent. Consequently,
$\\{L_{i}\\}_{i=0}^{m}$ is a basis for
$\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$ since its dimension is $m+1$
from Lemma 1.

∎

In addition to having a Lagrange basis, the RIFE space has an orthogonal basis
with respect to $(\cdot,\cdot)_{w,\check{I}}$ as stated in the following
theorem in which we also show that if a function
$\check{\varphi}\in\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$ is
orthogonal to $\mathcal{V}^{m-1}_{\check{\alpha},r}(\check{I})$ with respect
to $(\cdot,\cdot)_{w,\check{I}}$, then $\check{\varphi}$ has exactly $m$
distinct interior roots similar to the classical orthogonal polynomials.
Although the theorem holds for a general weight $w$, we will restrict our
attention to a piecewise constant function $w$:

$w(x)=\begin{cases}w_{-},&x\in\check{I}^{-},\\\
w_{+},&x\in\check{I}^{+},\end{cases}$ (22)

where $w_{\pm}$ are positive constants. The result of this theorem can also be
considered as a generalization for the theorem about the orthogonal IFE basis
described in [12] for elliptic interface problems.

######  Theorem 3.

Let ${\tilde{m}}\geq m\geq
1,\\{r_{k}\\}_{k=0}^{\tilde{m}}\subset\mathbb{R}_{+}$,
$\check{\alpha}\in(0,1)$, and let $w:\check{I}\to\mathbb{R}_{+}$ be defined as
in (22), then there is a non-zero
$\check{\varphi}\in\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$ such that

$\displaystyle(\check{\varphi},\check{\psi})_{w,\check{I}}=\int_{\check{I}}w(x)\check{\varphi}(x)\check{\psi}(x)\
dx=0,\quad\forall\check{\psi}\in\mathcal{V}^{m-1}_{\check{\alpha},r}(\check{I}).$
(23)

Furthermore, $\check{\varphi}$ has exactly $m$ distinct roots in the interior
of $\check{I}$.

###### Proof.

Existence is a classical result of linear algebra. The proof of the second
claim follows the same steps used for orthogonal polynomials: Note that
$\check{\varphi}$ has at least one root of odd multiplicity in the interior of
$\check{I}$ since
$(\check{\varphi},\mathcal{N}^{0}_{\check{\alpha},r})_{w,\check{I}}=0$.

Assume that $\check{\varphi}$ has $j<m$ distinct roots
$\\{\xi_{i}\\}_{i=1}^{j}$ of odd multiplicity in the interior of $\check{I}$.
Following the ideas in the proof of Theorem 2, we can show that there is
$\check{\psi}_{0}\in\mathcal{V}^{j}_{\check{\alpha},r}(\check{I})$ such that
$\check{\psi}_{0}(\xi_{i})=0$ for $1\leq i\leq j$.

Furthermore, all roots of $\check{\psi}_{0}$ are simple according to Theorem 1
since the sum of multiplicities cannot exceed $j$, and $\check{\psi}_{0}$
changes sign at these roots. This means that
$w\check{\varphi}\check{\psi}_{0}$ does not change sign on $\check{I}$. As a
consequence, $(\check{\varphi},\check{\psi}_{0})_{w,\check{I}}\neq 0$ which
contradicts the assumption $(\check{\varphi},\check{\psi})_{w,\check{I}}=0$
for all $\check{\psi}\in\mathcal{V}^{m-1}_{\check{\alpha},r}(\check{I})$ since
$\mathcal{V}^{j}_{\check{\alpha},r}(\check{I})\subseteq\mathcal{V}^{m-1}_{\check{\alpha},r}(\check{I})$.
∎

For every integer $m$ with ${\tilde{m}}\geq m\geq 1$, we use
$\mathcal{Q}^{m}_{\check{\alpha},w,r}(\check{I}),~{}m\geq 1$ to denote the
orthogonal complement of $\mathcal{V}^{m-1}_{\check{\alpha},r}(\check{I})$ in
$\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$ with respect to the weight
$w$, that is

$\mathcal{Q}^{m}_{\check{\alpha},w,r}(\check{I})=\left\\{\check{\varphi}\in\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})\mid(\check{\varphi},\check{\psi})_{w,\check{I}}=0,\
\forall\check{\psi}\in\mathcal{V}^{m-1}_{\check{\alpha},r}(\check{I})\right\\}.$

According to Theorem 3, one can see that
$\check{\varphi}\mapsto\sqrt{\check{\varphi}(0)^{2}+\check{\varphi}(1)^{2}}$
defines a norm on $\mathcal{Q}^{m}_{\check{\alpha},w,r}(\check{I})$ which is
one-dimensional. Thus, it is is equivalent to the $L^{2}$ norm and the
quantity
$\frac{\sqrt{\check{\varphi}(0)^{2}+\check{\varphi}(1)^{2}}}{\left\lVert\check{\varphi}\right\rVert_{0,\check{I}}}$
depends only on $\check{\alpha},w$ and $r$ (and not on the choice of
$\check{\varphi}\in\mathcal{Q}^{m}_{\check{\alpha},w,r}(\check{I})$).
Furthermore, The following lemma shows that the equivalence constant is
independent of the interface location. This result will be crucial later in
the analysis of Radau projections.

######  Lemma 4.

Let ${\tilde{m}}\geq m\geq
1,\\{r_{k}\\}_{k=0}^{\tilde{m}}\subset\mathbb{R}_{+}$,
$\check{\alpha}\in(0,1)$ and $w$ be defined as in (22) then, there exist
$C(m,w,r)$ and $\tilde{C}(m,w,r)>0$ independent of $\check{\alpha}$ such that
for every $\check{\varphi}\in\mathcal{Q}^{m}_{\check{\alpha},w,r}(\check{I})$,
we have

$\displaystyle\sqrt{\check{\varphi}(0)^{2}+\check{\varphi}(1)^{2}}\geq
C(m,w,r)\left\lVert\check{\varphi}\right\rVert_{0,\check{I}}\geq\tilde{C}(m,w,r)\left\lVert\check{\varphi}\right\rVert_{1,\check{I}}.$
(24)

###### Proof.

The inequality on the right follows from the inverse inequality (15) for the
IFE funcdtions. For a proof of the inequality on the left, see Appendix A.

∎

##  4 An immersed Bramble-Hilbert lemma and the approximation capabilities of
the LIFE space

In this section, we will develop a new version of Bramble-Hilbert lemma that
applies to functions in $\mathcal{H}^{m+1}_{\check{\alpha},r}(\check{I})$ and
its IFE counterpart. This lemma will serve as a fundamental tool for
investigating the approximation capability of IFE spaces. In the discussions
below, we will use $\mathbbm{1}_{B}$ for the indicator function of a set
$B\subset\mathbb{R}$ and we define
$w_{i}=r_{i}\mathbbm{1}_{\check{I}^{-}}+\mathbbm{1}_{\check{I}^{+}}$ for
$i=0,1,\dots,m$.

######  Theorem 4.

Let ${\tilde{m}}\geq m\geq
0,\\{r_{k}\\}_{k=0}^{\tilde{m}}\subset\mathbb{R}_{+}$,
$\check{\alpha}\in(0,1)$, and
$v\in\mathcal{H}^{m+1}_{\check{\alpha},r}(\check{I})$. Assume
$(w_{i},v^{(i)})_{\check{I}}=0$ for $i=0,1,\dots,m$. Then, there exists
$C(i,r)>0$ independent of $\check{\alpha}$ such that

$\displaystyle\left\lVert v\right\rVert_{i,\check{I}}\leq
C(i,r)|v|_{i,\check{I}},~{}i=0,1,\cdots,m+1.$ (25)

###### Proof.

Let $v\in\mathcal{H}^{m+1}_{\check{\alpha},r}(\check{I})$ and assume
$(w_{i},v^{(i)})_{\check{I}}=0$ for $i=0,1,\dots,m$. Because $w_{0}$ is such
that $w_{0}v$ is continuous and since $v_{\mid\check{I}^{\pm}}\in
H^{1}(\check{I}^{\pm})$, we have $w_{0}v\in H^{1}(\check{I})$. Therefore, for
any given $x,y\in\check{I}$, we have

$w_{0}(x)v(x)-w_{0}(y)u(y)=\int_{y}^{x}w_{0}(z)v^{\prime}(z)\ dz$

We integrate this identity on $\check{I}$ with respect to $x$ and use
$(w_{0},v^{(0)})_{\check{I}}=0$ to get

$-w_{0}(y)v(y)=\int_{0}^{1}\int_{y}^{x}w_{0}(z)v^{\prime}(z)\ dz\
dx,\quad\forall y\in\check{I}.$

Taking the absolute value and applying the Cauchy-Schwarz inequality, we get

$|v(y)|\leq\frac{1}{\min(1,r_{0})}|w_{0}(y)v(y)|\leq\frac{\max(1,r_{0})}{\min(1,r_{0})}|v|_{1,\check{I}}=\max\left(r_{0},r_{0}^{-1}\right)|v|_{1,\check{I}},$

which implies $\left\lVert
v\right\rVert_{0,\check{I}}\leq\max\left(r_{0},r_{0}^{-1}\right)|v|_{1,\check{I}}$.
Since $v^{\prime}\in\mathcal{H}^{m}_{\check{\alpha},\tau(r)}(\check{I})$,
where $\tau$ is the shift operator described in (17), we can use the same
reasoning to show that

$|v|_{1,\check{I}}\leq\max\left(r_{1},r_{1}^{-1}\right)|v|_{2,\check{I}}.$

Repeating the same arguments, we can obtain

$\displaystyle|v|_{i,\check{I}}\leq\max\left(r_{i},r_{i}^{-1}\right)|v|_{i+1,\check{I}},~{}~{}i=0,1,\cdots,m$
(26)

which leads to (25) with

$C(i,r)=\sqrt{1+\sum_{k=0}^{i}\prod_{j=k}^{i}\max\left(r_{j}^{2},r_{j}^{-2}\right)}.$

∎

######  Lemma 5.

Let ${\tilde{m}}\geq m\geq
0,\\{r_{k}\\}_{k=0}^{\tilde{m}}\subset\mathbb{R}_{+}$,
$\check{\alpha}\in(0,1)$ and
$v\in\mathcal{H}^{m+1}_{\check{\alpha},r}(\check{I})$. Then, there is a unique
$\check{\pi}^{m}_{\check{\alpha},r}v\in\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$
that satisfies

$\int_{\check{I}}w_{i}(x)\frac{d^{i}}{dx^{i}}\left(v(x)-\check{\pi}^{m}_{\check{\alpha},r}v(x)\right)\
dx=0,\qquad\forall i=0,1,\dots,m.$ (27)

###### Proof.

For $v\in\mathcal{H}^{m+1}_{\check{\alpha},r}(\check{I})$, to see that
$\check{\pi}^{m}_{\check{\alpha},r}v$ exists and is unique, we consider the
problem of finding
$\check{\varphi}\in\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$ such that

$(w_{i},\check{\varphi}^{(i)})_{\check{I}}=(w_{i},v^{(i)})_{\check{I}},\qquad\text{for
}i=0,1,\dots,m.$ (28)

By Lemma 1, we can express $\check{\varphi}$ in terms of the canonical basis

$\check{\varphi}=\sum_{j=0}^{m}c_{j}\mathcal{N}^{j}_{\check{\alpha},r}.$

Then, by (28), the coefficients $\mathbf{c}$ of $\check{\varphi}$ are
determined by the linear system $A\mathbf{c}=\mathbf{b}$, where $A=(A_{i,j})$
is a triangular matrix with
$A_{i,j}=(w_{i},(\mathcal{N}^{j}_{\check{\alpha},r})^{(i)})_{\check{I}},~{}0\leq
i,j\leq m$ and diagonal entries

$A_{i,i}=(w_{i},(\mathcal{N}^{i}_{\check{\alpha},r})^{(i)})_{\check{I}}=i!(h_{-}+r_{i}h_{+})\neq
0,~{}0\leq i\leq m.$

Therefore, $A$ is invertible and
$\check{\pi}^{m}_{\check{\alpha},r}v=\check{\varphi}$ is uniquely determined
by (27). ∎

We note that the mapping
$\check{\pi}^{m}_{\check{\alpha},r}:v\in\mathcal{H}^{m+1}_{\check{\alpha},r}(\check{I})\mapsto\check{\pi}^{m}_{\check{\alpha},r}v\in\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$
is linear because of the linearity of integration.

We now present an immersed version of the Bramble-Hilbert lemma [10] which can
be considered a generalization of the one-dimensional Bramble-Hilbert lemma in
the sense that if $r\equiv 1$, then, this immersed Bramble-Hilbert lemma
recovers the classical Bramble-Hilbert lemma.

######  Lemma 6.

Let ${\tilde{m}}\geq m\geq i\geq
0,\\{r_{k}\\}_{k=0}^{\tilde{m}}\subset\mathbb{R}_{+}$,
$\check{\alpha}\in(0,1)$. Assume
$\check{P}^{m}_{\check{\alpha},r}:\mathcal{H}^{m+1}_{\check{\alpha},r}(\check{I})\to\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$
is a linear map that satisfies the following two conditions

  1. 1.

$\check{P}^{m}_{\check{\alpha},r}$ is a projection on
$\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$ in the sense that

$\check{P}^{m}_{\check{\alpha},r}\check{\varphi}=\check{\varphi},\quad\forall\check{\varphi}\in\mathcal{V}^{m}_{\check{\alpha},r}(\check{I}).$
(29)

  2. 2.

There exists an integer $j$, $0\leq j\leq m+1$ such that
$\check{P}^{m}_{\check{\alpha},r}$ is bounded with respect to the norm
$\left\lVert\cdot\right\rVert_{j,\check{I}}$ as follows:

$\displaystyle\left\lVert\check{P}^{m}_{\check{\alpha},r}v\right\rVert_{i,\check{I}}\leq
C\left\lVert v\right\rVert_{j,\check{I}},\qquad\forall
v\in\mathcal{H}^{m+1}_{\check{\alpha},r}(\check{I}).$ (30)

Then, there exists $C(m,r)>0$ independent of $\check{\alpha}$, such that

$\displaystyle\left\lVert
v-\check{P}^{m}_{\check{\alpha},r}v\right\rVert_{i,\check{I}}\leq
C(m,r)\left(1+\left\lVert\check{P}^{m}_{\check{\alpha},r}\right\rVert_{i,j,\check{I}}\right)|v|_{m+1,\check{I}},\quad
v\in\mathcal{H}^{m+1}_{\check{\alpha},r}(\check{I}),$ (31)

where

$\displaystyle\left\lVert\check{P}^{m}_{\check{\alpha},r}\right\rVert_{i,j,\check{I}}=\sup\left\\{\left\lVert\check{P}^{m}_{\check{\alpha},r}v\right\rVert_{i,\check{I}}\mid
v\in\mathcal{H}^{m+1}_{\check{\alpha},r}(\check{I})\text{ and }\left\lVert
v\right\rVert_{j,\check{I}}=1\right\\}.$

###### Proof.

Since $\check{P}^{m}_{\check{\alpha},r}$ is a projection in the sense of (29),
we have
$\check{P}^{m}_{\check{\alpha},r}\check{\pi}^{m}_{\check{\alpha},r}v=\check{\pi}^{m}_{\check{\alpha},r}v$.
Using the triangle inequality and (30), we obtain

$\displaystyle\left\lVert
v-\check{P}^{m}_{\check{\alpha},r}v\right\rVert_{i,\check{I}}$
$\displaystyle\leq\left\lVert
v-\check{\pi}^{m}_{\check{\alpha},r}v\right\rVert_{i,\check{I}}+\left\lVert\check{P}^{m}_{\check{\alpha},r}\left(v-\check{\pi}^{m}_{\check{\alpha},r}v\right)\right\rVert_{i,\check{I}},$
$\displaystyle\leq\left(1+\left\lVert\check{P}^{m}_{\check{\alpha},r}\right\rVert_{i,j,\check{I}}\right)\left\lVert
v-\check{\pi}^{m}_{\check{\alpha},r}v\right\rVert_{j,\check{I}}\leq\left(1+\left\lVert\check{P}^{m}_{\check{\alpha},r}\right\rVert_{i,j,\check{I}}\right)\left\lVert
v-\check{\pi}^{m}_{\check{\alpha},r}v\right\rVert_{m+1,\check{I}},\qquad\forall
v\in\mathcal{H}^{m+1}_{\check{\alpha},r}(\check{I}).$

Then, applying Lemma 5 and Theorem 4 to the right hand side of the above
estimate leads to (31).

∎

Next, we extend the results of Lemma 6 to the physical interface element
$I_{k_{0}}=[x_{k_{0}-1},x_{k_{0}-1}+h]$. Following the tradition in finite
element analysis, for every function $\varphi$ defined on the interface
element $I_{k_{0}}$, we can map it to a function
$\mathcal{M}\varphi=\check{\varphi}$ defined on the reference interval
$\check{I}$ by the standard affine transformation:

$\mathcal{M}\varphi(\xi)=\check{\varphi}(\xi)=\varphi(x_{k_{0}-1}+h\xi),\quad\xi\in\check{I}=[0,1].$
(32)

Furthermore, given a mapping
$P^{m}_{\alpha,r}:\mathcal{H}^{m+1}_{\alpha,r}(I_{k_{0}})\to\mathcal{V}^{m}_{\alpha,r}(I_{k_{0}})$,
we can use this affine transformation to introduce a mapping
$\check{P}^{m}_{\check{\alpha},r}:\mathcal{H}^{m+1}_{\check{\alpha},r}(\check{I})\to\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$
such that

$(\check{P}^{m}_{\check{\alpha},r}\check{v})(\xi)=(P^{m}_{\alpha,r}v)(x_{k_{0}-1}+h\xi)=(P^{m}_{\alpha,r}v)(x)~{}~{}\text{with}~{}~{}\xi\in\check{I}~{}~{}\text{or}~{}~{}x=x_{k_{0}-1}+h\xi\in
I_{k_{0}}.$ (33)

It can be verified that the mappings $\mathcal{M},P^{m}_{\alpha,r}$ and
$\check{P}^{m}_{\check{\alpha},r}$ satisfy the following commutative diagram:

(34)

We now use the immersed Bramble-Hilbert lemma in the scaling argument to
obtain estimates for the projection error $v-P^{m}_{\alpha,r}v$.

######  Theorem 5.

Let ${\tilde{m}}\geq m\geq i\geq
0,\\{r_{k}\\}_{k=0}^{\tilde{m}}\subset\mathbb{R}_{+}$,
$\check{\alpha}\in(0,1)$. Assume
$P^{m}_{\alpha,r}:\mathcal{H}^{m+1}_{\alpha,r}(I_{k_{0}})\to\mathcal{V}^{m}_{\alpha,r}(I_{k_{0}})$
is a linear operator such that $\check{P}^{m}_{\check{\alpha},r}$ defined by
(33) satisfies the assumption of Lemma 6 for an integer $j$ with $0\leq j\leq
m+1$. Then, there exists $C(m,r)>0$ independent of $\alpha$ such that

$|v-P^{m}_{\alpha,r}v|_{i,I_{k_{0}}}\leq
Ch^{m+1-i}\left(1+\left\lVert\check{P}^{m}_{\check{\alpha},r}\right\rVert_{i,j,\check{I}}\right)|v|_{m+1,I_{k_{0}}},\quad
v\in\mathcal{H}_{\alpha,r}^{m+1}(I_{k_{0}}).$ (35)

###### Proof.

The proof follows the same argument as for the classical case $r_{k}=1,\
k=0,1,\dots,m$ [14]. We start by applying the change of variables $x\mapsto
h^{-1}(x-x_{k_{0}-1})$ to we obtain

$|v|_{m+1,I_{k_{0}}}=h^{-1}h^{-m-1}|\check{v}|_{m+1,\check{I}},\qquad|v-P^{m}_{\alpha,r}v|_{i,I_{k_{0}}}=h^{-1}h^{-i}|\check{v}-\check{P}^{m}_{\check{\alpha},r}\check{v}|_{i,\check{I}}.$
(36)

Next, we combine Lemma 6 and (36) to obtain

$\displaystyle|v-P^{m}_{\alpha,r}v|_{i,I_{k_{0}}}$
$\displaystyle=h^{-i-1}|\check{v}-\check{P}^{m}_{\check{\alpha},r}\check{v}|_{i,\check{I}}\leq
h^{-i-1}C(m,r)\left(1+\left\lVert\check{P}^{m}_{\check{\alpha},r}\right\rVert_{i,j,\check{I}}\right)|\check{v}|_{m+1,\check{I}}$
$\displaystyle=h^{-i-1}C(m,r)\left(1+\left\lVert\check{P}^{m}_{\check{\alpha},r}\right\rVert_{i,j,\check{I}}\right)h^{m+2}|v|_{m+1,I_{k_{0}}}$
$\displaystyle=C(m,r)h^{m+1-i}\left(1+\left\lVert\check{P}^{m}_{\check{\alpha},r}\right\rVert_{i,j,\check{I}}\right)|v|_{m+1,I_{k_{0}}}.$

∎

Nevertheless, the estimate (35) does not directly lead to the convergence of
$P^{m}_{\alpha,r}v$ to $v$ as $h\rightarrow 0$ unless we can show that
$\left\lVert\check{P}^{m}_{\check{\alpha},r}\right\rVert_{i,j,\check{I}}$ is
uniformly bounded with respect to $\check{\alpha}\in\check{I}$, and this can
be addressed by the uniform boundedness of $P^{m}_{\alpha,r}$ defined as
follows.

######  Definition 2.

Let ${\tilde{m}}\geq m\geq
0,\\{r_{k}\\}_{k=0}^{\tilde{m}}\subset\mathbb{R}_{+}$,
$\check{\alpha}\in\check{I}$, and let
$\\{\check{P}^{m}_{\check{\alpha},r}\\}_{0<\check{\alpha}<1}$ be a collection
of projections in the sense of (29) such that
$\check{P}^{m}_{\check{\alpha},r}:\mathcal{H}^{m+1}_{\check{\alpha},r}(\check{I})\to\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$.
We call $\\{\check{P}^{m}_{\check{\alpha},r}\\}_{0<\check{\alpha}<1}$ a
uniformly bounded collection of RIFE projections provided that there exists a
constant $C>0$ independent of $\check{\alpha}$ and an integer $j$ with $0\leq
j\leq m+1$ such that

$\left\lVert\check{P}^{m}_{\check{\alpha},r}v\right\rVert_{0,\check{I}}<C\left\lVert
v\right\rVert_{j,\check{I}},\ \forall
v\in\mathcal{H}^{m+1}_{\check{\alpha},r}(\check{I}),\qquad\forall\check{\alpha}\in(0,1),$
(37)

and the associated collection of maps
$\\{P^{m}_{\alpha,r}\\}_{\alpha\in\overset{\circ}{I}_{k_{0}}}$ defined in (33)
is called a uniformly bounded collection of LIFE projections.

######  Lemma 7.

Let ${\tilde{m}}\geq m\geq
0,\\{r_{k}\\}_{k=0}^{\tilde{m}}\subset\mathbb{R}_{+}$,
$\check{\alpha}\in\check{I}$. Assume
$\\{\check{P}^{m}_{\check{\alpha},r}\\}_{0<\check{\alpha}<1}$ is a uniformly
bounded collection of RIFE projections. Then, there exists a constant $C$
independent of $\check{\alpha}$ such that

$\displaystyle\left\lVert\check{P}^{m}_{\check{\alpha},r}v\right\rVert_{i,\check{I}}\leq
C\left\lVert v\right\rVert_{m+1,\check{I}},~{}\forall
v\in\mathcal{H}^{m+1}_{\check{\alpha},r}(\check{I}),~{}0\leq i\leq m+1.$ (38)

###### Proof.

Assume that $\\{\check{P}^{m}_{\check{\alpha},r}\\}_{0<\check{\alpha}<1}$ is a
uniformly bounded collection of RIFE projections, then,

$\displaystyle\left\lVert\check{P}^{m}_{\check{\alpha},r}v\right\rVert_{0,\check{I}}\leq
C\left\lVert v\right\rVert_{j,\check{I}}$

for an integer $j$ with $0\leq j\leq m+1$. By Lemma 3, we further have

$\displaystyle\left\lVert\check{P}^{m}_{\check{\alpha},r}v\right\rVert_{i,\check{I}}\leq
c(m,r)\left\lVert\check{P}^{m}_{\check{\alpha},r}v\right\rVert_{0,\check{I}}\leq
c(m,r)C\left\lVert v\right\rVert_{j,\check{I}}\leq c(m,r)C\left\lVert
v\right\rVert_{m+1,\check{I}},~{}~{}0\leq i\leq m+1$

which implies the uniform boundedness stated in (38).

∎

Now, we can derive an error bound for a collection of uniformly bounded LIFE
projections that implies convergence.

######  Theorem 6.

Let ${\tilde{m}}\geq m\geq
0,\\{r_{k}\\}_{k=0}^{\tilde{m}}\subset\mathbb{R}_{+}$,
$\check{\alpha}\in\check{I}$. Assume that
$\\{P^{m}_{\alpha,r}\\}_{\alpha\in\overset{\circ}{I}_{k_{0}}}$ is an uniformly
bounded collection of LIFE projections. Then, there exits a constant $C>0$
independent of $\alpha$ and $h$ such that

$|v-P^{m}_{\alpha,r}v|_{i,I_{k_{0}}}\leq
Ch^{m+1-i}|v|_{m+1,I_{k_{0}}},\quad\forall
v\in\mathcal{H}_{\alpha,r}^{m+1}(I_{k_{0}}),\quad\forall i=0,1,\dots,m.$ (39)

###### Proof.

By Lemma 7, we know that
$\\{P^{m}_{\alpha,r}\\}_{\alpha\in\overset{\circ}{I}_{k_{0}}}$ satisfies (30)
for $i=0,1,2,\dots,m$ with $j=m+1$. Consequently, we have

$\displaystyle\left\lVert\check{P}^{m}_{\check{\alpha},r}\right\rVert_{i,m+1,\check{I}}=\sup\left\\{\left\lVert\check{P}^{m}_{\check{\alpha},r}v\right\rVert_{i,\check{I}}\mid
v\in\mathcal{H}^{m+1}_{\check{\alpha},r}(\check{I})\text{ and }\left\lVert
v\right\rVert_{m+1,\check{I}}=1\right\\}\leq C.$

Then, applying these to (35) established in Theorem 5 yields (39).

∎

The simplest example of a uniformly bounded collection of LIFE projections is
the $L^{2}$ projection
$\check{\text{Pr}}_{\check{\alpha},r}^{m}:\mathcal{H}_{\check{\alpha},r}^{m+1}(\check{I})\to\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$
defined by

$\left(q,\check{\text{Pr}}_{\check{\alpha},r}^{m}v\right)_{\check{I}}=\left(q,v\right)_{\check{I}},\qquad\forall
q\in\mathcal{V}^{m}_{\check{\alpha},r}(\check{I}).$

Choosing $q=\check{\text{Pr}}_{\check{\alpha},r}^{m}v$, we get
$\left\lVert\check{\text{Pr}}_{\check{\alpha},r}^{m}v\right\rVert_{0,\check{I}}=\left\lVert
v\right\rVert_{0,\check{I}}\leq\left\lVert v\right\rVert_{m+1,\check{I}}$. By
Lemma 7,
$\\{\check{\text{Pr}}_{\check{\alpha},r}^{m}\\}_{\alpha\in\overset{\circ}{I}_{k_{0}}}$
is an uniformly bounded collection of projections. Consequently, by Theorem 6,
we can obtain the following optimal approximation capability of the LIFE
space.

######  Corollary 1.

Let ${\tilde{m}}\geq m\geq
0,\\{r_{k}\\}_{k=0}^{\tilde{m}}\subset\mathbb{R}_{+}$,
$\check{\alpha}\in(0,1)$. Then, there exists a constant $C>0$ independent of
$\alpha$ and $h$ such that

$\min_{q\in\mathcal{V}^{m}_{\alpha,r}(I_{k_{0}})}|q-v|_{0,I_{k_{0}}}\leq
Ch^{m+1}|v|_{m+1,I_{k_{0}}},\quad\forall
v\in\mathcal{H}_{\alpha,r}^{m+1}(I_{k_{0}}).$ (40)

##  5 Analysis of an IFE-DG method for a class of hyperbolic systems with
discontinuous coefficients

In this section, we employ the results of Section 3 and Section 4 to analyse
an IFE-DG method for the acoustic interface problem (1). To our knowledge, the
analysis of such method for hyperbolic systems has so far not been considered
in the literature unlike the IFE methods for elliptic problems. The main
challenge that time-dependent problems present is the plethora of possible
jump coefficients. For instance, the jump coefficients $r^{p}_{k},r^{v}_{k}$
described in (2) for the acoustic interface problem are given by:

$r^{p}_{2k}=r^{v}_{2k}=\left(\frac{c_{-}}{c_{+}}\right)^{2k},\qquad
r^{p}_{2k+1}=\frac{\rho_{+}}{\rho_{-}}\left(\frac{c_{-}}{c_{+}}\right)^{2k},\qquad
r^{v}_{2k+1}=\frac{\rho_{-}}{\rho_{+}}\left(\frac{c_{-}}{c_{+}}\right)^{2k+2},\quad
k=0,1,\dots$ (41)

The nature of the jump coefficients in (41) makes the study of this particular
IFE space extremely tedious as observed in [32]. Fortunately, the theory
developed in Section 3 and Section 4 is general and applies to any choice of
positive jump coefficients.

###  5.1 Problem statement and preliminary results

Let $I=(a,b)$ be a bounded interval containing $\alpha$, and let
$\rho_{\pm},c_{\pm}$ be positive constants describing the density and the
sound speed in $I^{\pm}$, respectively. Now, we consider the acoustic
interface problem on $I$

$\mathbf{u}_{t}(x,t)+A(x)\mathbf{u}_{x}(x,t)=0,\qquad x\in
I\backslash\\{\alpha\\},\qquad t>0,$ (42a) where $\mathbf{u}=(p,u)^{T}$ is the
pressure-velocity couple and $A_{\mid
I^{\pm}}=A_{\pm}=\begin{pmatrix}0&\rho_{\pm}c_{\pm}^{2}\\\
\rho_{\pm}^{-1}&0\end{pmatrix}.$ (42b) The matrices $A_{\pm}$ can be
decomposed as $A_{\pm}=P_{\pm}\Lambda_{\pm}P_{\pm}^{-1}$, where
$\Lambda_{\pm}=\mathrm{diag}(-c_{\pm},c_{\pm})$. Using this eigen-
decomposition, we define
$A_{\pm}^{+}=P_{\pm}\mathrm{diag}(0,c_{\pm})P_{\pm}^{-1}$,
$A_{\pm}^{-}=P_{\pm}\mathrm{diag}(-c_{\pm},0)P_{\pm}^{-1}$, and
$|A_{\pm}|=P_{\pm}\mathrm{diag}(c_{\pm},c_{\pm})P_{\pm}^{-1}$ to be the
positive part,the negative part, and the absolute value of $A_{\pm}$,
respectively. The acoustic interface problem that we are considering here is
subject to the following homogeneous inflow boundary conditions
$A_{-}^{+}\mathbf{u}(a,t)=A_{+}^{-}\mathbf{u}(b,t)=0,\qquad t\geq 0,$ (42c)
initial conditions $\mathbf{u}(x,0)=\mathbf{u}_{0}(x),x\in I,$ (42d) and
interface condition $\mathbf{u}(\alpha^{-},t)=\mathbf{u}(\alpha^{+},t),\qquad
t\geq 0.$ (42e)

In the remainder of this section, let
$S_{\pm}=\mathrm{diag}(\rho_{\pm}^{-1}c_{\pm}^{-2},\rho_{\pm})$ and
$S(x)=S_{\pm}$ if $x\in I^{\pm}$, then

$S_{\pm}A_{\pm}=\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}=\tilde{A}.$ (43)

Now, we can multiply (42a) by $S$ and write the acoustic interface problem as

$S(x)\mathbf{u}_{t}(x,t)+\tilde{A}\mathbf{u}_{x}(x,t)=0,\qquad x\in
I\backslash\\{\alpha\\},\qquad t>0.$ (44)

Lombard and Piraux [30], and Moon [32] have shown that by successively
differentiating
$\left\llbracket\mathbf{u}(\cdot,t)\right\rrbracket_{\alpha}=0$, where
$\left\llbracket\cdot\right\rrbracket$ is the jump, we obtain

$\left\llbracket
A(\cdot)^{k}\mathbf{u}(\cdot,t)\right\rrbracket_{\alpha}=0\Longleftrightarrow\frac{\partial^{k}}{\partial
x^{k}}\mathbf{u}(\alpha^{+},t)=R_{k}\frac{\partial^{k}}{\partial
x^{k}}\mathbf{u}(\alpha^{-},t),\quad R_{k}=A_{+}^{-k}A_{-}^{k},\
k=0,1,\dots,m.$

Since $R_{k}$ is diagonal (see part (a) of Lemma 8), the condition
$\mathbf{u}^{(k)}(\alpha^{+},t)=R_{k}\mathbf{u}^{(k)}(\alpha^{-},t)$ is
equivalent to

$\frac{\partial^{k}}{\partial
x^{k}}p(\alpha^{+},t)=r^{p}_{k}\frac{\partial^{k}}{\partial
x^{k}}p(\alpha^{-},t),\qquad\frac{\partial^{k}}{\partial
x^{k}}u(\alpha^{+},t)=r^{u}_{k}\frac{\partial^{k}}{\partial
x^{k}}u(\alpha^{-},t),$ (45)

where $r^{p}_{k}$ and $r^{u}_{k}$ are defined in (41). These decoupled
interface conditions make the results obtained previously about the
approximation capabilities of the LIFE space directly applicable to vector
functions in the product spaces

$\mathbb{H}^{m+1}_{\alpha,\mathbf{r}}(I_{k_{0}})=\mathcal{H}^{m+1}_{\alpha,r^{p}}(I_{k_{0}})\times\mathcal{H}^{m+1}_{\alpha,r^{u}}(I_{k_{0}}),\quad\mathbb{V}^{m}_{\alpha,\mathbf{r}}(I_{k_{0}})=\mathcal{V}^{m}_{\alpha,r^{p}}(I_{k_{0}})\times\mathcal{V}^{m}_{\alpha,r^{u}}(I_{k_{0}}),\quad\mathbb{W}^{m}_{\alpha,\mathbf{r}}(\mathcal{T}_{h})=W^{m}_{\alpha,r^{p}}(\mathcal{T}_{h})\times
W^{m}_{\alpha,r^{u}}(\mathcal{T}_{h}),$

where $\mathbf{r}=(r^{p},r^{u})$. Now, we define the following bilinear forms

$M(\mathbf{w},\mathbf{v})=\sum_{k=1}^{N}\left(S\mathbf{v},\mathbf{w}\right)_{I_{k}}$
(46a)
$B(\mathbf{w},\mathbf{v})=\sum_{k=1}^{N}\left(\mathbf{v}^{\prime},\tilde{A}\mathbf{w}\right)_{I_{k}}+\sum_{k=0}^{N}\left\llbracket\mathbf{v}\right\rrbracket_{x_{k}}^{T}S(x_{k})\hat{\mathbf{w}}(x_{k}),$
(46b)

where the numerical flux
$\hat{\mathbf{w}}(x_{k})=A(x_{k})^{+}\mathbf{w}(x_{k}^{-})+A(x_{k})^{-}\mathbf{w}(x_{k}^{+})$
at the interior nodes. At the boundary, we have
$\hat{\mathbf{w}}(x_{N})=A(x_{N})^{+}\mathbf{w}(x_{N}^{-})$,
$\hat{\mathbf{w}}(x_{0})=A(x_{0})^{-}\mathbf{w}(x_{0}^{+})$,
$\left\llbracket\mathbf{w}\right\rrbracket_{x_{N}}=\mathbf{w}(x_{N}^{-})$ and
$\left\llbracket\mathbf{w}\right\rrbracket_{x_{0}}=-\mathbf{w}(x_{0}^{+})$.

Now we define the immersed DG formulation as: Find $\mathbf{u}_{h}\in
C^{1}\left([0,T],\mathbb{W}^{m}_{\alpha,\mathbf{r}}(\mathcal{T}_{h})\right)$
such that

$\displaystyle(\mathbf{u}_{h}(\cdot,0),\mathbf{v}_{h})_{I}=(\mathbf{u}_{0},\mathbf{v}_{h})_{I},\qquad$
$\displaystyle\forall\mathbf{v}_{h}\in\mathbb{W}^{m}_{\alpha,\mathbf{r}}(\mathcal{T}_{h}),$
(47a) $\displaystyle
M(\mathbf{u}_{h,t}(\cdot,t),\mathbf{v}_{h})=B(\mathbf{u}_{h}(\cdot,t),\mathbf{v}_{h}),\qquad$
$\displaystyle\forall\mathbf{v}_{h}\in\mathbb{W}^{m}_{\alpha,\mathbf{r}}(\mathcal{T}_{h}),$
(47b)

We note that the discrete weak form (47) and the discrete space
$\mathbb{W}^{m}_{\alpha,\mathbf{r}}(\mathcal{T}_{h})$ are identical to the
ones described in the IDPGFE formulation in [32].

Next, we will go through some basic properties of the matrices $S_{\pm}$ and
$A_{\pm}$, these properties will be used later in the proof the $L^{2}$
stability in Lemma 9, in the analysis of the immersed Radau projection and in
the convergence estimate.

######  Lemma 8.

Let $A_{\pm}$ be the matrices defined in (42b) and let
$S_{\pm}=\mathrm{diag}(\rho_{\pm}^{-1}c_{\pm}^{-1},\rho_{\pm})$, then

  1. (a)

For any integer $k\geq 0$, the matrix $A_{+}^{-k}A_{-}^{k}$ is diagonal with
positive entries.

  2. (b)

Let $s\in\\{+,-\\}$, then there is an invertible matrix $P_{s}$ such that
$A_{s}=P_{s}\mathrm{diag}(-c_{s},c_{s})P_{s}^{-1}$ and
$S_{s}=P_{s}^{-T}P_{s}^{-1}$.

  3. (c)

Let $s\in\\{+,-\\}$, then the matrices $S_{s}A_{s}^{+},\ S_{s}A_{s}^{-}$ and
$S_{s}|A_{s}|$ are symmetric. Furthermore, $S_{s}A^{+}_{s}$ is positive semi-
definite, $S_{s}A^{-}_{s}$ is negative semi-definite and $S_{s}|A_{s}|$ is
positive definite.

  4. (d)

Let $s,\tilde{s}\in\\{+,-\\}$, and let $\mathbf{w}\in\mathbb{R}^{2}$, then

$\left(\left\lVert
A^{\tilde{s}}_{s}\mathbf{w}\right\rVert^{2}+\left|\mathbf{w}^{T}S_{s}A^{\tilde{s}^{\prime}}_{s}\mathbf{w}\right|=0\right)\Longrightarrow\mathbf{w}=0.$
(48)

where $\tilde{s}^{\prime}$ is dual of $\tilde{s}$ defined at the beginning of
Section 3, and $\left\lVert\cdot\right\rVert$ is Euclidean norm.

  5. (e)

Let $s\in\\{+,-\\}$. Then, there is a constant $C(\rho_{s},c_{s})>0$ such that

$\mathbf{w}^{T}S_{s}A_{s}^{+}\mathbf{w}-\mathbf{w}^{T}S_{s}A_{s}^{-}\mathbf{w}\geq
C(\rho_{s},c_{s})\left\lVert\mathbf{w}\right\rVert^{2},\qquad\forall\mathbf{w}\in\mathbb{R}^{2}.$
(49)

###### Proof.

  1. (a)

We have $A_{\pm}^{2}=c_{\pm}^{2}\mathrm{Id}_{2}$, where $\mathrm{Id}_{2}$ is
the the $2\times 2$ identity matrix. Therefore,

$A_{\pm}^{2k}=c_{\pm}^{2k}\mathrm{Id}_{2},\qquad
A_{\pm}^{2k+1}=c_{\pm}^{2k}A_{\pm},\qquad k=0,1,\dots$ (50)

Using (50), we immediately obtain
$A_{+}^{-2k}A_{-}^{2k}=\left(\frac{c_{-}}{c_{+}}\right)^{2k}\mathrm{Id}_{2}$
and
$A_{+}^{-2k-1}A_{-}^{2k+1}=\left(\frac{c_{-}}{c_{+}}\right)^{2k}A_{+}^{-1}A_{-}$.
Finally, by direct computation, we have
$A_{+}^{-1}A_{-}=\mathrm{diag}\left(\frac{\rho_{+}}{\rho_{-}},\frac{\rho_{-}c_{-}^{2}}{\rho_{+}c_{+}^{2}}\right)$.
Hence, $A_{+}^{-k}A_{-}^{k}=\mathrm{diag}(r^{p}_{k},r^{u}_{k})$, where
$r^{p}_{k}$ and $r^{u}_{k}$are defined in (41).

  2. (b)

Let

$P_{s}=\frac{1}{\sqrt{2\rho_{s}}}\begin{pmatrix}-c_{s}\rho_{s}&c_{s}\rho_{s}\\\
1&1\end{pmatrix},$ (51)

then, by a simple computation, we can show that $S_{s}=P_{s}^{-T}P_{s}^{-1}$
and $A_{s}=P_{s}\mathrm{diag}(-c_{s},c_{s})P_{s}^{-1}$.

  3. (c)

We have $S_{s}A_{s}^{+}=P_{s}^{-T}\mathrm{diag}(0,c_{s})P_{s}^{-1}$, where
$P_{s}$ is defined in (51). Therefore, $S_{s}A_{s}^{+}$ is a symmetric semi-
positive definite matrix. The other two claims can be proven similarly.

  4. (d)

We will only consider the case $\tilde{s}=+$ here, the other case can be
proven similarly. Consider a vector $\mathbf{w}\in\mathbb{R}^{2}$ that
satisfies

$\left\lVert
A^{+}_{s}\mathbf{w}\right\rVert^{2}+\left|\mathbf{w}^{T}S_{s}A^{-}_{s}\mathbf{w}\right|=0.$
(52)

Now, let $\tilde{\mathbf{w}}=P_{s}\mathbf{w}$ where $P_{s}$ is defined in
(51), then (52) can be written as

$\left\lVert
P_{s}\mathrm{diag}(0,c_{s})\tilde{\mathbf{w}}\right\rVert^{2}+\left\lVert\mathrm{diag}(-c_{s},0)\tilde{\mathbf{w}}\right\rVert^{2}=0,$

Since both norms are non-negative, we have
$\mathrm{diag}(-c_{s},0)\tilde{\mathbf{w}}=0$ and
$P_{s}\mathrm{diag}(0,c_{s})\tilde{\mathbf{w}}=0$. $P_{s}$ being invertible,
we get $\tilde{\mathbf{w}}=0$. Consequently,
$\mathbf{w}=P_{s}^{-1}\tilde{\mathbf{w}}=0$.

  5. (e)

We have by direct computation

$\mathbf{w}^{T}S_{s}(A_{s}^{+}-A_{s}^{-})\mathbf{w}=\mathbf{w}^{T}\begin{pmatrix}\rho_{s}^{-1}c_{s}^{-1}&0\\\
0&\rho_{s}c_{s}\end{pmatrix}\mathbf{w}\geq\min(\rho_{s}c_{s},\rho_{s}^{-1}c_{s}^{-1})\left\lVert\mathbf{w}\right\rVert.$

∎

######  Lemma 9.

Let $\mathbf{u}$ be a solution to (42), and let
$\epsilon(t)=\frac{1}{2}\left(\mathbf{u}(\cdot,t),S(\cdot)\mathbf{u}(\cdot,t)\right)_{0,I}^{2}$,
then

$\epsilon^{\prime}(t)\leq 0,\qquad t\geq 0.$

###### Proof.

By multiplying (44) by $\mathbf{u}^{T}$ and integrating on $I^{\pm}$, we
obtain

$\int_{I}\mathbf{u}(x,t)^{T}S\mathbf{u}_{t}(x,t)+\mathbf{u}(x,t)^{T}\tilde{A}\mathbf{u}_{x}(x,t)\
dx=0.$

The matrices $S$ and $\tilde{A}$ are symmetric. Therefore, we rewrite the
previous equation as

$\epsilon^{\prime}(t)+\sum_{s=+,-}\int_{I^{s}}\frac{\partial}{\partial
x}\mathbf{u}(x,t)^{T}\tilde{A}\mathbf{u}(x,t))dx=0.$

Since $\mathbf{u}$ is continuous at $\alpha$ (from (42e)), we have

$\epsilon^{\prime}(t)+\mathbf{u}(b,t)^{T}\tilde{A}\mathbf{u}(b,t)-\mathbf{u}(a,t)^{T}\tilde{A}\mathbf{u}(a,t)=0.$
(53)

Now, we can rewrite the term $\mathbf{u}(b,t)^{T}\tilde{A}\mathbf{u}(b,t)$ as

$\mathbf{u}(b,t)^{T}\tilde{A}\mathbf{u}(b,t)=\mathbf{u}(b,t)^{T}S_{+}A_{+}\mathbf{u}(b,t)=\mathbf{u}(b,t)^{T}S_{+}A_{+}^{+}\mathbf{u}(b,t)+\mathbf{u}(b,t)^{T}S_{+}A^{-}_{+}\mathbf{u}(b,t)=\mathbf{u}(b,t)^{T}S_{+}A_{+}^{+}\mathbf{u}(b,t),$

where the last equality follows from the boundary condition (42c). Since
$SA^{+}$ is symmetric semi-positive definite (see part (c) of Lemma 8), we
conclude that $\mathbf{u}(b,t)^{T}\tilde{A}\mathbf{u}(b,t)\geq 0$. Similarly,
we have $\mathbf{u}(a,t)^{T}\tilde{A}\mathbf{u}(a,t)\leq 0$. Therefore,

$\epsilon^{\prime}(t)\leq 0.$

∎

The previous lemma shows that $\epsilon(t)$, interpreted as the energy of the
system, is decreasing. This is to be expected since the boundary conditions in
(42c) are dissipative (see [9]). Furthermore, if we let
$\epsilon_{h}(t)=\frac{1}{2}M(\mathbf{u}_{h}(\cdot,t),\mathbf{u}_{h}(\cdot,t))$
be the discrete energy, then

$\epsilon_{h}^{\prime}(t)=B(\mathbf{u}_{h},\mathbf{u}_{h})=\frac{-1}{2}\sum_{k=0}^{N}\left\llbracket\mathbf{u}_{h}\right\rrbracket^{T}_{x_{k}}S(x_{k})|A(x_{k})|\left\llbracket\mathbf{u}_{h}\right\rrbracket_{x_{k}}\leq
0.$ (54)

The proof of (54) follows the same steps as the scalar case described in [15].

###  5.2 The immersed Radau projection and the convergence analysis

We denote by
$\mathcal{R}\mathbf{u}\in\mathbb{W}^{m}_{\alpha,\mathbf{r}}(\mathcal{T}_{h})$
the global Gauss-Radau projection defined as

$B(\mathbf{u}-\mathcal{R}\mathbf{u},\mathbf{v}_{h})=0,\qquad\forall\mathbf{v}_{h}\in\mathbb{W}^{m}_{\alpha,\mathbf{r}}(\mathcal{T}_{h}).$
(55)

Although $\mathcal{R}\mathbf{u}$ is a global projection, it can be constructed
on each element independently, the construction of
$(\mathcal{R}\mathbf{u})_{\mid I_{k}}$ where $k\neq k_{0}$ can be found in [1,
36] for the scalar case and can be generalized easily to systems. On the
interface element, we define the local immersed Radau projection (IRP)
operator
$\Pi^{m}_{\alpha,\mathbf{r}}:\mathbb{H}^{m+1}_{\alpha,\mathbf{r}}(I_{k_{0}})\to\mathbb{V}^{m}_{\alpha,\mathbf{r}}(I_{k_{0}})$
using (34) as

$\Pi^{m}_{\alpha,\mathbf{r}}=\mathcal{M}^{-1}\circ\check{\Pi}^{m}_{\check{\alpha},\mathbf{r}}\circ\mathcal{M},$

where
$\check{\Pi}^{m}_{\check{\alpha},\mathbf{r}}:\mathbb{H}^{m+1}_{\check{\alpha},\mathbf{r}}(\check{I})\to\mathbb{V}^{m}_{\check{\alpha},\mathbf{r}}(\check{I})$
is called the reference IRP operator and it is defined as the solution to the
following system:

$\displaystyle[left={\empheqlbrace}]A^{-}_{-}\check{\Pi}^{m}_{\check{\alpha},\mathbf{r}}\check{\mathbf{u}}(0$
$\displaystyle=A^{-}_{-}\check{\mathbf{u}}(0),$ (56a) $\displaystyle
A^{+}_{+}\check{\Pi}^{m}_{\check{\alpha},\mathbf{r}}\check{\mathbf{u}}(1)$
$\displaystyle=A^{+}_{+}\check{\mathbf{u}}(1),$ (56b)
$\displaystyle\left(\tilde{A}\mathbf{v}^{\prime},\check{\Pi}^{m}_{\check{\alpha},\mathbf{r}}\check{\mathbf{u}}\right)_{\check{I}}$
$\displaystyle=\left(\tilde{A}\mathbf{v}^{\prime},\check{\mathbf{u}}\right)_{\check{I}},\qquad\forall\mathbf{v}\in\mathbb{V}^{m}_{\check{\alpha},\mathbf{r}}(\check{I}).$
(56c)

Next, we will go through some basic properties of the IRP to prove that the
IRP is well defined and is uniformly bounded on the RIFE space
$\mathbb{V}^{m}_{\check{\alpha},\mathbf{r}}(\check{I})$. From there, we can
show that the IRP error on the LIFE space
$\mathbb{V}^{m}_{\alpha,\mathbf{r}}(I_{k_{0}})$ decays at an optimal rate of
$O(h^{m+1})$ under mesh refinement.

######  Lemma 10.

Let $A$ be the matrix function defined in (42b) and let
$\mathbf{p}\in\mathbb{V}^{m}_{\check{\alpha},\mathbf{r}}(\check{I})$, then
$A\mathbf{p}^{\prime}\in\mathbb{V}^{m-1}_{\check{\alpha},\mathbf{r}}(\check{I})$.
Furthermore the map

$\displaystyle G:\mathbb{V}^{m}_{\check{\alpha},\mathbf{r}}(\check{I})$
$\displaystyle\to\mathbb{V}^{m-1}_{\check{\alpha},\mathbf{r}}(\check{I})$
$\displaystyle\mathbf{p}$ $\displaystyle\mapsto A\mathbf{p}^{\prime}$

is surjective.

###### Proof.

Let $\mathbf{p}\in\mathbb{V}^{m}_{\check{\alpha},\mathbf{r}}(\check{I})$ and
let $\tilde{\mathbf{p}}=A\mathbf{p}^{\prime}$, then for a fixed
$k\in\\{0,1,\dots,m-1\\}$, we have

$\displaystyle\tilde{\mathbf{p}}^{(k)}(\check{\alpha}^{+})$
$\displaystyle=A_{+}\mathbf{p}^{(k+1)}(\check{\alpha}^{+})$ (Using
$\tilde{\mathbf{p}}=A\mathbf{p}^{\prime}$ )
$\displaystyle=A_{+}A_{+}^{-k-1}A_{-}^{k+1}\mathbf{p}^{(k+1)}(\check{\alpha}^{-})$
(By construction of $\mathbb{V}^{m}_{\check{\alpha},\mathbf{r}}(\check{I})$)
$\displaystyle=A_{+}A_{+}^{-k-1}A_{-}^{k+1}A_{-}^{-1}\tilde{\mathbf{p}}^{(k)}(\check{\alpha}^{-}),$
(Using $\mathbf{p}^{\prime}=A^{-1}\tilde{\mathbf{p}}$)
$\displaystyle=A_{+}^{-k}A_{-}^{k}\tilde{\mathbf{p}}^{(k)}(\check{\alpha}^{-}).$
(56be)

Since (56be) holds for every $k=0,1,\dots,m-1$, we conclude that
$\tilde{\mathbf{p}}\in\mathbb{V}^{m-1}_{\check{\alpha},\mathbf{r}}(\check{I})$.

Now, we show that $G$ is surjective, by the rank-nullity theorem, it suffices
to prove that $\dim\mathrm{ker}(G)$ is $2$ since
$\dim\mathbb{V}^{m}_{\check{\alpha},\mathbf{r}}(\check{I})-\dim\mathbb{V}^{m-1}_{\check{\alpha},\mathbf{r}}(\check{I})=2$.
Let $\mathbf{p}\in\mathrm{ker}(G)$, then $A\mathbf{p}^{\prime}=0$, since $A$
is invertible, we get $\mathbf{p}^{\prime}=0$ which implies that
$\mathbf{p}\in\mathbb{V}^{0}_{\check{\alpha},\mathbf{r}}(\check{I})$. This
shows that
$\dim\mathrm{ker}(G)=\dim\mathbb{V}^{0}_{\check{\alpha},\mathbf{r}}(\check{I})=2$.
∎

Following the definition of $G$, we can re-write (56c) as

$(SG(\mathbf{v}),\check{\Pi}^{m}_{\check{\alpha},\mathbf{r}}\check{\mathbf{u}})_{\check{I}}=(SG(\mathbf{v}),\check{\mathbf{u}})_{\check{I}},\qquad\forall\mathbf{v}\in\mathbb{V}^{m}_{\check{\alpha},\mathbf{r}}(\check{I}).$

For convenience, we will write
$(SG(\mathbf{v}),\check{\Pi}^{m}_{\check{\alpha},\mathbf{r}}\check{\mathbf{u}})_{\check{I}}=(G(\mathbf{v}),\check{\Pi}^{m}_{\check{\alpha},\mathbf{r}}\check{\mathbf{u}})_{S,\check{I}}$.
Now, since $G$ maps $\mathbb{V}^{m}_{\check{\alpha},\mathbf{r}}(\check{I})$
onto $\mathbb{V}^{m-1}_{\check{\alpha},\mathbf{r}}(\check{I})$, we can express
the condition (56c) as

$(\mathbf{v},\check{\Pi}^{m}_{\check{\alpha},\mathbf{r}}\check{\mathbf{u}})_{S,\check{I}}=(\mathbf{v},\check{\mathbf{u}})_{S,\check{I}},\qquad\forall\mathbf{v}\in\mathbb{V}^{m-1}_{\check{\alpha},\mathbf{r}}(\check{I}).$
(56bf)

######  Theorem 7.

The system (56) admits exactly one solution.

###### Proof.

First, we prove that the system admits at most one solution, for that, we only
need to show that if $\check{\mathbf{u}}=0$, then
$\check{\Pi}^{m}_{\check{\alpha},\mathbf{r}}\check{\mathbf{u}}=0$. For
simplicity, let $\mathbf{q}=(q_{1},q_{2})^{T}$ be the solution to (56) with
$\check{\mathbf{u}}=0$, and let $r^{(1)}=r^{p}$ and $r^{(2)}=r^{u}$, then by
(56bf), we have

$(v,q_{i})_{w_{i},\check{I}}=0,\ \forall
v\in\mathcal{V}^{m-1}_{\check{\alpha},r^{(i)}}(\check{I}),\text{ where
}w_{i}=S_{i,i},\qquad i=1,2,$ (56bg)

which is equivalent to
$q_{i}\in\mathcal{Q}^{m}_{\check{\alpha},w_{i},r^{(i)}}(\check{I})$. On the
other hand, we have

$\left(\tilde{A}\mathbf{q}^{\prime},\mathbf{q}\right)_{\check{I}}=\frac{1}{2}\left(\mathbf{q}(1)^{T}S_{+}A_{+}\mathbf{q}(1)-\mathbf{q}(0)^{T}S_{-}A_{-}\mathbf{q}(0)\right)=0.$
(56bh)

From (56a) and (56b), we have
$A^{+}_{+}\mathbf{q}(1)=A^{-}_{-}\mathbf{q}(0)=0$, then
$A_{+}\mathbf{q}(1)=A_{+}^{-}\mathbf{q}(1)$ and
$A_{-}\mathbf{q}(0)=A_{-}^{+}\mathbf{q}(0)$. Therefore, the equation (56bh)
becomes

$\frac{1}{2}\left(\mathbf{q}(1)^{T}S_{+}A_{+}^{-}\mathbf{q}(1)-\mathbf{q}(0)^{T}S_{-}A_{-}^{+}\mathbf{q}(0)\right)=0.$

Now, by Lemma 8 part (c), the quantities
$\mathbf{q}(1)^{T}S_{+}A_{+}^{-}\mathbf{q}(1)$ and
$-\mathbf{q}(0)^{T}S_{-}A_{-}^{+}\mathbf{q}(0)$ are non-positive, then

$\mathbf{q}(1)^{T}S_{+}A_{+}^{-}\mathbf{q}(1)=\mathbf{q}(0)^{T}S_{-}A_{-}^{+}\mathbf{q}(0)=0,$

Furthermore, by (56a) and (56b), we have

$\left\lVert
A_{+}^{+}\mathbf{q}(1)\right\rVert^{2}+|\mathbf{q}(1)^{T}S_{+}A_{+}^{-}\mathbf{q}(1)|=\left\lVert
A_{-}^{-}\mathbf{q}(0)\right\rVert^{2}+|\mathbf{q}(0)^{T}S_{-}A_{-}^{+}\mathbf{q}(0)|=0.$

At this point, we use Lemma 8 part (d) to conclude that
$\mathbf{q}(1)=\mathbf{q}(0)=0$. Therefore, $q_{i}$ are orthogonal IFE
functions (as shown in (56bg)) that vanish on the boundary. By Theorem 3, we
conclude that $q_{i}\equiv 0$ for $i=1,2$. Equivalently, $\mathbf{q}\equiv 0$.

To finalize the proof, we only need to show that (56) can be written as a
square system. Let
$A_{\pm}=P_{\pm}\mathrm{diag}(-c_{\pm},c_{\pm})P_{\pm}^{-1}$ be an eigen-
decomposition of $A_{\pm}$. Then, (56) can be written as

$\begin{cases}\left(P_{-}^{-1}\check{\Pi}^{m}_{\check{\alpha},\mathbf{r}}\check{\mathbf{u}}(0)\right)_{1}&=\left(P_{-}^{-1}\check{\mathbf{u}}(0)\right)_{1}\\\
\left(P_{+}^{-1}\check{\Pi}^{m}_{\check{\alpha},\mathbf{r}}\check{\mathbf{u}}(1)\right)_{2}&=\left(P_{+}^{-1}\check{\mathbf{u}}(1)\right)_{2},\\\
\left(\mathcal{N}^{j}_{\check{\alpha},r^{p}},(\check{\Pi}^{m}_{\check{\alpha},\mathbf{r}}\check{\mathbf{u}})_{i}\right)_{S_{11},\check{I}}&=\left(\mathcal{N}^{j}_{\check{\alpha},r^{p}},\check{p}\right)_{S_{22},\check{I}},\qquad\
1\leq j\leq m-1,\\\
\left(\mathcal{N}^{j}_{\check{\alpha},r^{u}},(\check{\Pi}^{m}_{\check{\alpha},\mathbf{r}}\check{\mathbf{u}})_{i}\right)_{S_{22},\check{I}}&=\left(\mathcal{N}^{j}_{\check{\alpha},r^{u}},\check{u}\right)_{S_{22},\check{I}},\qquad\
1\leq j\leq m-1\end{cases}$ (56bi)

which is a system of $2(m+1)$ equations with $2(m+1)$ variables. Since the
homogeneous system admits at most one solution, we conclude that (56) has
exactly one solution.

∎

Next, we show that
$\\{\check{\Pi}^{m}_{\check{\alpha},\mathbf{r}}\\}_{0<\check{\alpha}<1}$ is
uniformly bounded. First, let
$\mathbf{p}\in\mathbb{V}^{m-1}_{\check{\alpha},\mathbf{r}}(\check{I})$ be the
solution to the following symmetric positive definite system

$\left(\mathbf{v},\mathbf{p}\right)_{S,\check{I}}=\left(\mathbf{v},\check{\mathbf{u}}\right)_{\check{I}},\
\forall\mathbf{v}\in\mathbb{V}^{m-1}_{\check{\alpha},\mathbf{r}}(\check{I}),$
(56bj)

and let
$\mathbf{q}=\check{\Pi}^{m}_{\check{\alpha},\mathbf{r}}\check{\mathbf{u}}-\mathbf{p}$,
then by (56c) and (56bf), we have

$\left(\tilde{A}\mathbf{v}^{\prime},\mathbf{q}\right)_{\check{I}}=\left(\tilde{A}\mathbf{v}^{\prime},\check{\Pi}^{m}_{\check{\alpha},\mathbf{r}}\check{\mathbf{u}}-\mathbf{p}\right)_{\check{I}}=0,\qquad\forall\mathbf{v}\in\mathbb{V}^{m}_{\check{\alpha},\mathbf{r}}(\check{I}),$
(56bk)

which can be written as

$\left(\mathbf{v},\mathbf{q}\right)_{S,\check{I}}=0,\qquad\forall\mathbf{v}\in\mathbb{V}^{m-1}_{\check{\alpha},\mathbf{r}}(\check{I}).$

Thus,
$\mathbf{q}\in\mathbb{Q}^{m}_{\check{\alpha},S,\mathbf{r}}(\check{I})=\mathcal{Q}^{m}_{\check{\alpha},S_{11},r^{p}}(\check{I})\times\mathcal{Q}^{m}_{\check{\alpha},S_{22},r^{u}}(\check{I})$.
Additionally, by (56a) and (56b), we have

$\begin{cases}A^{-}_{-}\mathbf{q}(0)=A^{-}_{-}\left(\check{\mathbf{u}}(0)-\mathbf{p}(0)\right),\\\
A^{+}_{+}\mathbf{q}(1)=A^{+}_{+}\left(\check{\mathbf{u}}(1)-\mathbf{p}(1)\right).\end{cases}$
(56bl)

In the next two lemmas, we prove that
$\left\lVert\mathbf{p}\right\rVert_{0,\check{I}}$ and
$\left\lVert\mathbf{q}\right\rVert_{0,\check{I}}$ is bounded by some
appropriate norms of
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\check{\mathbf{u}}}$
independently of $\check{\alpha}$. Both lemmas will be used later in Theorem 8
to prove that
$\\{\check{\Pi}^{m}_{\check{\alpha},\mathbf{r}}\\}_{0<\check{\alpha}<1}$ is a
uniformly bounded collection of RIFE projections.

######  Lemma 11.

Let $\check{\mathbf{u}}\in\mathbb{H}^{m+1}_{\check{\alpha},r}(\check{I})$ and
$\mathbf{p}\in\mathbb{V}^{m-1}_{\check{\alpha},\mathbf{r}}(\check{I})$ defined
by (56bj), then there is $C(\rho,c)>0$ independent of $\check{\alpha}$ such
that $\left\lVert\mathbf{p}\right\rVert_{0,\check{I}}\leq
C(\rho,c)\left\lVert\check{\mathbf{u}}\right\rVert_{0,\check{I}}$.

###### Proof.

Let $\mathbf{v}=\mathbf{p}$ in (56bj), then
$\left\lVert\mathbf{p}^{T}S\mathbf{p}\right\rVert_{0,\check{I}}=(\mathbf{p},\check{\mathbf{u}})_{0,\check{I}}\leq\left\lVert\mathbf{p}\right\rVert_{0,\check{I}}\left\lVert\check{\mathbf{u}}\right\rVert_{0,\check{I}}$.
On the other hand, by construction of $S$, we have
$\left\lVert\mathbf{p}^{T}S\mathbf{p}\right\rVert_{0,\check{I}}\geq
C(\rho,c)\left\lVert\mathbf{p}\right\rVert_{0,\check{I}}$, where
$C(\rho,c)=\min(\rho_{-}^{-1}c_{-}^{-2},\rho_{+}^{-1}c_{+}^{-2},\rho_{-},\rho_{+})$.
Therefore,

$\left\lVert\mathbf{p}\right\rVert_{0,\check{I}}\leq
C(\rho,c)\left\lVert\check{\mathbf{u}}\right\rVert_{0,\check{I}}.$

∎

######  Lemma 12.

Let $\check{\mathbf{u}}\in\mathbb{H}^{m+1}_{\check{\alpha},r}(\check{I})$ and
let $\mathbf{p}\in\mathbb{V}^{m-1}_{\check{\alpha},\mathbf{r}}(\check{I})$
defined by (56bj). Then, there is $C(\rho,c,m)>0$ independent of
$\check{\alpha}$ such that

$\left\lVert\check{\Pi}^{m}_{\check{\alpha},\mathbf{r}}\check{\mathbf{u}}-\mathbf{p}\right\rVert_{0,\check{I}}\leq
C(\rho,c,m)\left\lVert\check{\mathbf{u}}\right\rVert_{1,\check{I}}$

###### Proof.

Let
$\mathbf{q}=(q_{1},q_{2})=\check{\Pi}^{m}_{\check{\alpha},\mathbf{r}}\check{\mathbf{u}}-\mathbf{p}$,
then by (56bk), we have
$(\tilde{A}\mathbf{q}^{\prime},\mathbf{q})_{\check{I}}=0$ which is equivalent
to

$\mathbf{q}(1)^{T}\tilde{A}\mathbf{q}(1)-\mathbf{q}(0)^{T}\tilde{A}\mathbf{q}(0)=0$
(56bm)

By decomposing $A_{\pm}=A^{+}_{\pm}+A^{-}_{\pm}$ and using the definition of
$\tilde{A}$ in (43), we can split (56bm) as

$\mathbf{q}(1)^{T}S_{+}A_{+}^{+}\mathbf{q}(1)+\mathbf{q}(1)^{T}S_{+}A_{+}^{-}\mathbf{q}(1)-\mathbf{q}(0)^{T}S_{-}A_{-}^{+}\mathbf{q}(0)-\mathbf{q}(0)^{T}S_{-}A_{-}^{-}\mathbf{q}(0)=0.$
(56bn)

Now, from (49), we have

$\mathbf{q}(0)^{T}S_{-}A_{-}^{+}\mathbf{q}(0)-\mathbf{q}(0)^{T}S_{-}A_{-}^{-}\mathbf{q}(0)\geq
C_{1}(\rho,c)\left\lVert\mathbf{q}(0)\right\rVert^{2},$ (56boa)
$\mathbf{q}(1)^{T}S_{+}A_{+}^{+}\mathbf{q}(1)-\mathbf{q}(1)^{T}S_{+}A_{+}^{-}\mathbf{q}(1)\geq
C_{1}(\rho,c)\left\lVert\mathbf{q}(1)\right\rVert^{2}.$ (56bob)

Next, we sum (56bn), (56boa) and (56bob) to obtain

$\mathbf{q}(1)S_{+}A_{+}^{+}\mathbf{q}(1)-\mathbf{q}(0)S_{-}A_{-}^{-}\mathbf{q}(0)\geq\frac{1}{2}C_{1}(\rho,c)\left(\left\lVert\mathbf{q}(0)\right\rVert^{2}+\left\lVert\mathbf{q}(1)\right\rVert^{2}\right)$
(56bp)

We substitute (56bl) in (56bp) to obtain

$\mathbf{q}(1)S_{+}A_{+}^{+}(\check{\mathbf{u}}(1)-\mathbf{p}(1))-\mathbf{q}(0)S_{-}A_{-}^{-}(\check{\mathbf{u}}(0)-\mathbf{p}(0))\geq\frac{1}{2}C_{1}(\rho,c)\left(\left\lVert\mathbf{q}(0)\right\rVert^{2}+\left\lVert\mathbf{q}(1)\right\rVert^{2}\right)$
(56bq)

Now, we will bound the left hand side from above. First, we have

$\mathbf{q}(1)S_{+}A_{+}^{+}(\check{\mathbf{u}}(1)-\mathbf{p}(1))-\mathbf{q}(0)S_{-}A_{-}^{-}(\check{\mathbf{u}}(0)-\mathbf{p}(0))\leq
C_{2}(\rho,c)\left(\left\lVert\mathbf{q}(1)\right\rVert\left\lVert\check{\mathbf{u}}(1)-\mathbf{p}(1)\right\rVert+\left\lVert\mathbf{q}(0)\right\rVert\left\lVert\check{\mathbf{u}}(0)-\mathbf{p}(0)\right\rVert\right)$
(56br)

Since $\check{\mathbf{u}}-\mathbf{p}\in(H^{1}(\check{I}))^{2}$, there is
$C_{3}>0$ such that

$\max(\left\lVert\check{\mathbf{u}}(0)-\mathbf{p}(0)\right\rVert,\left\lVert\check{\mathbf{u}}(1)-\mathbf{p}(1)\right\rVert)\leq
C_{3}\left(\left\lVert\check{\mathbf{u}}\right\rVert_{1,\check{I}}+\left\lVert\mathbf{p}\right\rVert_{1,\check{I}}\right).$
(56bs)

By applying the inverse inequality (15) and Lemma 11 to
$\left\lVert\mathbf{p}\right\rVert_{1,\check{I}}$, we obtain

$\displaystyle\max(\left\lVert\check{\mathbf{u}}(0)-\mathbf{p}(0)\right\rVert,\left\lVert\check{\mathbf{u}}(1)-\mathbf{p}(1)\right\rVert)$
$\displaystyle\leq
C_{4}(\rho,c,m)\left(\left\lVert\check{\mathbf{u}}\right\rVert_{1,\check{I}}+\left\lVert\mathbf{p}\right\rVert_{0,\check{I}}\right),$
$\displaystyle\leq
C_{5}(\rho,c,m)\left\lVert\check{\mathbf{u}}\right\rVert_{1,\check{I}}$ (56bt)

Now, we substitute (56bt) and (56br) back into (56bq) and use the inequality
$a^{2}+b^{2}\geq\frac{1}{2}(a+b)^{2}$ to obtain

$\left(\left\lVert\mathbf{q}(0)\right\rVert+\left\lVert\mathbf{q}(1)\right\rVert\right)\left\lVert\check{\mathbf{u}}\right\rVert_{1,\check{I}}\geq
C_{6}(\rho,c,m)\left(\left\lVert\mathbf{q}(0)\right\rVert+\left\lVert\mathbf{q}(1)\right\rVert\right)^{2},$

which yields

$\left\lVert\mathbf{q}(0)\right\rVert+\left\lVert\mathbf{q}(1)\right\rVert\leq
C_{7}(\rho,c,m)\left\lVert\check{\mathbf{u}}\right\rVert_{1,\check{I}}.$
(56bu)

To finish the proof, we recall that
$\mathbf{q}\in\mathcal{Q}^{m}_{\check{\alpha},S_{11},r^{p}}(\check{I})\times\mathcal{Q}^{m}_{\check{\alpha},S_{22},r^{u}}(\check{I})$.
Therefore, by Lemma 4 and some elementary algebraic manipulations, we have

$\displaystyle\left\lVert\mathbf{q}\right\rVert_{0,\check{I}}$
$\displaystyle=\sqrt{\left\lVert
q_{1}\right\rVert_{0,\check{I}}^{2}+\left\lVert
q_{2}\right\rVert_{0,\check{I}}^{2}}$ (By defintion) $\displaystyle\leq
C_{7}(\rho,c,m)\sqrt{q_{1}(0)^{2}+q_{1}(1)^{2}+q_{2}(0)^{2}+q_{2}(1)^{2}},$
(Using Lemma 4) $\displaystyle\leq
C_{8}(\rho,c,m)\left(\left\lVert\mathbf{q}(0)\right\rVert+\left\lVert\mathbf{q}(1)\right\rVert\right)$
$\displaystyle\left\lVert\mathbf{q}\right\rVert_{0,\check{I}}$
$\displaystyle\leq
C_{9}(\rho,c,m)\left\lVert\check{\mathbf{u}}\right\rVert_{1,\check{I}},$
(using (56bu))

which is the desired result. ∎

By combining Lemma 11 and Lemma 12, we can show that the norm of
$\check{\Pi}^{m}_{\check{\alpha},\mathbf{r}}\check{\mathbf{u}}$ can be bounded
by a norm of $\check{\mathbf{u}}$ independently of $\check{\alpha}$ as
described in the following theorem. We note that
$\check{\Pi}^{m}_{\check{\alpha},\mathbf{r}}$ maps
$\mathbb{H}^{m+1}_{\check{\alpha},\mathbf{r}}(\check{I})$ to
$\mathbb{V}^{m}_{\check{\alpha},\mathbf{r}}(\check{I})$. Nevertheless, we
shall call $\check{\Pi}^{m}_{\check{\alpha},\mathbf{r}}$ a RIFE projection
since the results from Section 4 in the scalar case apply directly to the
vector case here.

######  Theorem 8.

Let $m\geq 1$ and let
$\check{\mathbf{u}}\in\mathbb{H}^{m+1}_{\check{\alpha},\mathbf{r}}(\check{I})$.
Then, there is $C(\rho,c,m)>0$ independent of $\check{\alpha}$ such that

$\left\lVert\check{\Pi}^{m}_{\check{\alpha},\mathbf{r}}\check{\mathbf{u}}\right\rVert_{0,\check{I}}\leq
C(\rho,c,m)\left\lVert\check{\mathbf{u}}\right\rVert_{1,\check{I}}$

That is,
$\\{\check{\Pi}^{m}_{\check{\alpha},\mathbf{r}}\\}_{0<\check{\alpha}<1}$ is a
uniformly bounded collection of RIFE projections.

###### Proof.

By definition of $\mathbf{p}$ and $\mathbf{q}$, we have

$\left\lVert\check{\Pi}^{m}_{\check{\alpha},\mathbf{r}}\check{\mathbf{u}}\right\rVert_{0,\check{I}}\leq\left\lVert\mathbf{p}\right\rVert_{0,\check{I}}+\left\lVert\mathbf{q}\right\rVert_{0,\check{I}},$

Which, using Lemma 11 and Lemma 12, leads to

$\left\lVert\check{\Pi}^{m}_{\check{\alpha},\mathbf{r}}\check{\mathbf{u}}\right\rVert_{0,\check{I}}\leq
C(\rho,c,m)\left\lVert\check{\mathbf{u}}\right\rVert_{1,\check{I}},$

where $C(\rho,c,m)>0$ is independent of $\check{\alpha}$. ∎

######  Corollary 2.

Let $\mathbf{u}\in\mathbb{H}^{m+1}_{\alpha,\mathbf{r}}(I_{k_{0}})$, then there
is $C(S_{\pm},A_{\pm},m)>0$ independent of $\alpha$ and $h$ such that

$\left|\mathbf{u}-\Pi^{m}_{\check{\alpha},\mathbf{r}}\mathbf{u}\right|_{i,I_{k_{0}}}<C(S_{\pm},A_{\pm},m)h^{m+1-i}|\mathbf{u}|_{m+1,I_{k_{0}}},\qquad
0\leq i\leq m.$

###### Proof.

Let $\check{\mathbf{u}}=(\check{p},\check{u})^{T}=\mathcal{M}\mathbf{u}$ and
let
$\check{\bm{\pi}}^{m}_{\check{\alpha},\mathbf{r}}\check{u}=(\check{\pi}^{m}_{\check{\alpha},r^{p}}\check{p},\check{\pi}^{m}_{\check{\alpha},r^{u}}\check{u})^{T}$,
where
$\check{\pi}^{m}_{\check{\alpha},r^{p}}\check{p},\check{\pi}^{m}_{\check{\alpha},r^{u}}$
are defined in Lemma 5. Then by Theorem 4, we have

$|\check{\mathbf{u}}-\check{\bm{\pi}}^{m}_{\check{\alpha},\mathbf{r}}\check{\mathbf{u}}|_{i,\check{I}}\leq
C(\rho,c,m)|\check{\mathbf{u}}|_{m+1,\check{I}},\qquad i=0,1,\dots,m+1.$
(56bv)

On the other hand, we have

$\displaystyle\left|\mathbf{u}-\Pi^{m}_{\check{\alpha},\mathbf{r}}\mathbf{u}\right|_{i,I_{k_{0}}}$
$\displaystyle=h^{1-i}\left|\check{\mathbf{u}}-\check{\Pi}^{m}_{\check{\alpha},\mathbf{r}}\check{\mathbf{u}}\right|_{i,\check{I}}$
$\displaystyle\leq
h^{1-i}\left(\left|\check{\Pi}^{m}_{\check{\alpha},\mathbf{r}}\left(\check{\mathbf{u}}-\check{\bm{\pi}}^{m}_{\check{\alpha},\mathbf{r}}\check{\mathbf{u}}\right)\right|_{i,\check{I}}+\left|\check{\mathbf{u}}-\check{\bm{\pi}}^{m}_{\check{\alpha},\mathbf{r}}\check{\mathbf{u}}\right|_{i,\check{I}}\right)$
$\displaystyle\leq
C_{1}(\rho,c,m)h^{1-i}\left(\left|\check{\Pi}^{m}_{\check{\alpha},\mathbf{r}}\left(\check{\mathbf{u}}-\check{\bm{\pi}}^{m}_{\check{\alpha},\mathbf{r}}\check{\mathbf{u}}\right)\right|_{0,\check{I}}+\left|\check{\mathbf{u}}-\check{\bm{\pi}}^{m}_{\check{\alpha},\mathbf{r}}\check{\mathbf{u}}\right|_{i,\check{I}}\right)$
(Using Lemma 3) $\displaystyle\leq
C_{2}(\rho,c,m)h^{1-i}\left(\left\lVert\check{\mathbf{u}}-\check{\bm{\pi}}^{m}_{\check{\alpha},\mathbf{r}}\check{\mathbf{u}}\right\rVert_{1,\check{I}}+\left|\check{\mathbf{u}}-\check{\bm{\pi}}^{m}_{\check{\alpha},\mathbf{r}}\check{\mathbf{u}}\right|_{i,\check{I}}\right)$
(Using Theorem 8) $\displaystyle\leq
C_{3}(\rho,c,m)h^{1-i}\left\lVert\check{\mathbf{u}}-\check{\bm{\pi}}^{m}_{\check{\alpha},\mathbf{r}}\check{\mathbf{u}}\right\rVert_{m+1,\check{I}}$
$\displaystyle\leq C_{4}(\rho,c,m)h^{1-i}|\check{\mathbf{u}}|_{m+1,\check{I}}$
(From (56bv) )
$\displaystyle=C_{4}(\rho,c,m)h^{m+1-i}|\mathbf{u}|_{m+1,I_{k_{0}}}.$

∎

By summing over all elements, we get a similar bound for the global Radau
projection $\mathcal{R}\mathbf{u}$ with a function
$\mathbf{u}\in\mathbb{H}^{m+1}_{\alpha,\mathbf{r}}(I)$

$\left\lVert\mathbf{u}-\mathcal{R}\mathbf{u}\right\rVert_{i,I}<C(S_{\pm},A_{\pm},m)h^{m+1-i}|\mathbf{u}|_{m+1,I},\qquad
0\leq i\leq m.$ (56bw)

######  Theorem 9.

Let $\mathbf{u}$ be the solution of problem (42) and let
$\mathbf{u}_{h}\in\mathbb{W}^{m}_{\alpha,\mathbf{r}}(I)$ be the solution of
(47). If $\mathbf{u}\in C([0,T];\mathbb{H}^{m+2}_{\alpha,\mathbf{r}}(I))$,
then there is $C>0$ independent of $h$ and $\alpha$ such that

$\left\lVert\mathbf{u}(\cdot,T)-\mathbf{u}_{h}(\cdot,T)\right\rVert_{0,I}\leq
Ch^{m+1}\left(|\mathbf{u}_{0}|_{m+1,I}+|\mathbf{u}(\cdot,T)|_{m+1,I}+T\max_{0\leq
t\leq T}|\mathbf{u}(\cdot,t)|_{m+2,I}\right),\qquad T>0.$

###### Proof.

Our proof follows the usual methodology used for the non-interface problem
(see [15]). We first note that
$\mathcal{R}\mathbf{u}_{t}=\frac{d}{dt}\mathcal{R}\mathbf{u}$ and split the
error $\mathbf{e}=\mathbf{u}_{h}-\mathbf{u}$ as

$\mathbf{e}=\mathbf{z}-\mathbf{g},\qquad\mathbf{z}=\mathbf{u}-\mathcal{R}\mathbf{u},\quad\mathbf{g}=\mathbf{u}_{h}-\mathcal{R}\mathbf{u}.$

It follows from the definition of $\mathcal{R}$ in (55) that

$B(\mathbf{g},\mathbf{g})=B(\mathbf{g}-\mathbf{z},\mathbf{g})=B(\mathbf{u}_{h}-\mathbf{u},\mathbf{g})=B(\mathbf{e},\mathbf{g}).$
(56bx)

By combining (56bx), (47) and (54), we get

$\displaystyle\left(S\mathbf{z}_{t}(\cdot,t),\mathbf{g}(\cdot,t)\right)_{I}$
$\displaystyle=\left(S\mathbf{g}_{t}(\cdot,t),\mathbf{g}(\cdot,t)\right)_{I}-\left(S\mathbf{e}_{t}(\cdot,t),\mathbf{g}(\cdot,t)\right)_{I},$
$\displaystyle=\frac{1}{2}\frac{d}{dt}\left\lVert\sqrt{S}\mathbf{g}(\cdot,t)\right\rVert_{0,I}^{2}-B(\mathbf{e}(\cdot,t),\mathbf{g}(\cdot,t)),$
$\displaystyle=\frac{1}{2}\frac{d}{dt}\left\lVert\sqrt{S}\mathbf{g}(\cdot,t)\right\rVert_{0,I}^{2}-B(\mathbf{g}(\cdot,t),\mathbf{g}(\cdot,t)),$
$\displaystyle=\frac{1}{2}\frac{d}{dt}\left\lVert\sqrt{S}\mathbf{g}(\cdot,t)\right\rVert_{0,I}^{2}+\sigma(t),$
(56by)

where $\sigma(t)\geq 0$ by (54). Let
$\kappa(t)=\left\lVert\sqrt{S}\mathbf{g}(\cdot,t)\right\rVert_{0,I}$, then by
Cauchy-Schwarz inequality,

$\left(S\mathbf{z}_{t}(\cdot,t),\mathbf{g}(\cdot,t)\right)_{I}\leq\left\lVert\mathbf{z}_{t}(\cdot,t)\right\rVert_{0,I}\kappa(t).$
(56bz)

Following the ideas of the proof of Lemma 10, we can show that
$\mathbf{u}_{t}(\cdot,t)=-A\mathbf{u}_{x}(\cdot,t)\in\mathbb{H}^{m+1}_{\alpha,\mathbf{r}}(I)$
since $\mathbf{u}(\cdot,t)\in\mathbb{H}^{m+2}_{\alpha,\mathbf{r}}(I)$.
Therefore, by (56bw), there is $C$ independent of $h$ and $\alpha$ such that

$\left\lVert\mathbf{z}_{t}(\cdot,t)\right\rVert_{0,I}\leq
Ch^{m+1}|\mathbf{u}(\cdot,t)|_{m+2,I}.$ (56ca)

Now, we use (56ca) , (56bz) and integrate (56by) on $[0,T]$ to get

$\frac{1}{2}\kappa(T)^{2}-\frac{1}{2}\kappa(0)^{2}+\sigma(t)\leq
Ch^{m+1}\int_{0}^{T}\kappa(s)|\mathbf{u}(\cdot,s)|_{m+2,I}\ ds,$ (56cb)

Using a generalized version of Gronwall’s inequality (see [8, p. 24]), we get
the following bound on $\kappa(T)$

$\displaystyle\kappa(T)$
$\displaystyle\leq\kappa(0)+Ch^{m+1}\int_{0}^{T}|\mathbf{u}(\cdot,s)|_{m+2,I}\
ds,$ (56cc) $\displaystyle\leq\kappa(0)+Ch^{m+1}T\max_{0\leq t\leq
T}|\mathbf{u}(\cdot,t)|_{m+2,I}.$ (56cd)

We also have

$\kappa(0)=\left\lVert\sqrt{S}\left(\mathbf{u}_{h}(\cdot,0)-\mathcal{R}\mathbf{u}_{0}\right)\right\rVert_{0,I}\leq\left\lVert\sqrt{S}\left(\mathbf{u}_{h}(\cdot,0)-\mathbf{u}_{0}\right)\right\rVert_{0,I}+\left\lVert\sqrt{S}\left(\mathbf{u}_{0}-\mathcal{R}\mathbf{u}_{0}\right)\right\rVert_{0,I}\leq
Ch^{m+1}|\mathbf{u}_{0}|_{m+1,I}.$ (56ce)

We substitute (56ce) into (56cd), to obtain

$\kappa(T)=\left\lVert\sqrt{S}\mathbf{g}(\cdot,T)\right\rVert_{0,I}\leq
Ch^{m+1}\left(|\mathbf{u}_{0}|_{m+1,I}+T\max_{0\leq t\leq
T}|\mathbf{u}(\cdot,t)|_{m+2,I}\right).$

To finalize the proof, we use the triangle inequality

$\left\lVert\mathbf{e}(\cdot,T)\right\rVert_{0,I}\leq\left\lVert\mathbf{z}(\cdot,T)\right\rVert_{0,I}+\left\lVert\mathbf{g}(\cdot,T)\right\rVert_{0,I}\leq
Ch^{m+1}\left(|\mathbf{u}_{0}|_{m+1,I}+|\mathbf{u}(\cdot,T)|_{m+1,I}+T\max_{0\leq
t\leq T}|\mathbf{u}(\cdot,t)|_{m+2,I}\right).$

∎

##  6 Novel proofs for results already established in the literature

In this section, for demonstrating the versatility of the immersed scaling
argument established in Section 3 and Section 4, we redo the error estimation
for two IFE methods in the literature. One of them is the IFE space for an
elliptic interface problem [5], and the other one is the IFE space for an
interface problem of the Euler-Bernoulli Beam [24]. We note that the
approximation capability for these IFE spaces were already analyzed, but with
complex and lengthy procedures. Our discussions here is to demonstrate that
similar error bounds for the optimal approximation capability of these
different types of IFE spaces can be readily derived by the unified immersed
scaling argument.

###  6.1 The $m$-th degree IFE space for an elliptic interface problem

In this subsection, we consider the $m$-th degree IFE space developed in [5]
for solving the following interface problem:

$\begin{cases}-\beta(x)u^{\prime\prime}(x)=f(x),\
x\in(a,\alpha)\cup(\alpha,b)\\\
u(a)=u(b)=0,\end{cases}\qquad\beta(x)=\begin{cases}\beta^{-}>0,&x\in(a,\alpha),\\\
\beta^{+}>0,&x\in(\alpha,b),\end{cases}\qquad[u]_{\alpha}=[\beta
u^{\prime}]_{\alpha}=0.$ (56cf)

Assume that $f$ is in $C^{m-1}(I)$ which implies that the solution
$u\in\mathcal{H}^{m+1}_{\alpha,r}(I)$ with

$r_{0}=1,\quad\text{ and }\quad r_{i}=\frac{\beta^{-}}{\beta^{+}}\ \text{ for
}\ i=1,2,\dots,m.$ (56cg)

The discussion in Section 2 suggests the following IFE space for this elliptic
interface problem:

$\displaystyle Z_{\alpha,r}^{m}(\mathcal{T}_{h})=H^{1}_{0}(I)\cap
W_{\alpha,r}^{m}(\mathcal{T}_{h})$ (56ch)

which coincides with the one developed in [5] based on the extended jump
conditions where it was proved, by an elementary but complicated multi-point
Taylor expansion technique, to have the optimal approximation capability with
respect to $m$-th degree polynomials employed in this IFE space. We now
reanalyze this IFE space by the immersed scaling argument.

The continuity of functions in the IFE space suggests to consider the
following immersed Lobatto projection
$\mathscr{L}^{m}_{\alpha,r}:\mathcal{H}^{m+1}_{\alpha,r}(I_{k_{0}})\to\mathcal{V}^{m}_{\alpha,r}(I_{k_{0}})$
defined by

$\begin{cases}\mathscr{L}^{m}_{\alpha,r}u(x_{k_{0}-1})=u(x_{k_{0}-1}),\\\
\mathscr{L}^{m}_{\alpha,r}u(x_{k_{0}})=u(x_{k_{0}}),\\\
\left(\mathscr{L}^{m}_{\alpha,r}u,v_{h}\right)_{w,I_{k_{0}}}=\left(u,v_{h}\right)_{w,I_{k_{0}}},\quad\forall
v_{h}\in\mathcal{V}^{m-2}_{\alpha,\tau^{2}(r)}(I_{k_{0}}),\end{cases}\qquad
w(x)=\begin{cases}r_{1},&x\in I_{k_{0}}^{-},\\\ 1,&x\in
I_{k_{0}}^{+},\end{cases}$ (56ci)

where $\tau^{2}=\tau\circ\tau$ and $\tau$ is the shift operator defined in
(17). The related reference immersed Lobatto projection
$\check{\mathscr{L}}^{m}_{\check{\alpha},r}:\mathcal{H}^{m+1}_{\check{\alpha},r}(\check{I})\to\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$
is defined by the diagram (34), that is,
$\check{\mathscr{L}}^{m}_{\check{\alpha},r}\check{u}=\mathscr{L}^{m}_{\alpha,r}u$
where $\check{u}=\mathcal{M}v$.

For simplicity, let
$\tilde{u}=\check{\mathscr{L}}^{m}_{\check{\alpha},r}\check{u}$ for a given
$\check{u}\in\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$ and note that the
system (56ci) is a square system of $m+1$ equations since the last line can be
written as $m-1$ equations. Therefore, we only need to show that if
$\check{u}\equiv 0$ then $\tilde{u}\equiv 0$ to prove that
$\check{\mathscr{L}}^{m}_{\check{\alpha},r}$ is well defined.

######  Lemma 13.

The reference immersed Lobatto projection
$\check{\mathscr{L}}^{m}_{\check{\alpha},r}$ is well defined.

###### Proof.

Let $\check{u}\equiv 0$, we will show that
$\tilde{u}=\check{\mathscr{L}}^{m}_{\check{\alpha},r}\check{u}\equiv 0$. We
have

$\tilde{u}(0)=\tilde{u}(1)=0,\qquad\left(\tilde{u},v_{h}\right)_{\check{w},\check{I}}=0,\
\forall v\in\mathcal{V}^{m-2}_{\check{\alpha},\tau^{2}(r)}(\check{I}),$

where $\check{w}=\mathcal{M}w$. Using (17),
$\tilde{u}^{\prime\prime}\in\mathcal{V}^{m-2}_{\check{\alpha},\tau^{2}(r)}(\check{I})$,
then

$\displaystyle 0=\int_{0}^{1}w(x)\tilde{u}(x)\tilde{u}^{\prime\prime}(x)\ dx$
$\displaystyle=r_{1}\int_{0}^{\check{\alpha}}\tilde{u}(x)\tilde{u}^{\prime\prime}(x)\
dx+\int_{\check{\alpha}}^{1}\tilde{u}(x)\tilde{u}^{\prime\prime}(x)\ dx$
$\displaystyle=\tilde{u}(\check{\alpha})\left[r_{1}\tilde{u}^{\prime}(\check{\alpha}^{-})-\tilde{u}^{\prime}(\check{\alpha}^{+})\right]-\int_{0}^{1}w(x)[\tilde{u}^{\prime}(x)]^{2}\
dx$ $\displaystyle 0$
$\displaystyle=-\int_{0}^{1}w(x)[\tilde{u}^{\prime}(x)]^{2}\ dx,$

which implies that $\tilde{u}$ is zero since $\tilde{u}(0)=\tilde{u}(1)=0$. ∎

Next, we will show that
$\\{\check{\mathscr{L}}^{m}_{\check{\alpha},r}\\}_{0\leq\check{\alpha}<1}$ is
a uniformly bounded collection of RIFE projections in the following lemma.

######  Lemma 14.

There is a constant $C(\beta^{+},\beta^{-},m)>0$ independent of
$\check{\alpha}$ such that the following estimate holds for every
$\check{u}\in\mathcal{H}^{m+1}_{\check{\alpha},r}(\check{I})$

$\left\lVert\tilde{u}\right\rVert_{0,\check{I}}\leq
C(\beta^{+},\beta^{-},m)\left\lVert\check{u}\right\rVert_{1,\check{I}}.$

###### Proof.

We write $\tilde{u}$ as $\tilde{u}=q_{1}+q_{2}$, where
$q_{1}\in\mathcal{V}^{1}_{\check{\alpha},r}(\check{I})$ such that

$q_{1}(0)=\check{u}(0),\quad q_{1}(1)=\check{u}(1),$

and
$q_{2}=\check{\mathscr{L}}^{m}_{\check{\alpha},r}(\check{u}-q_{1})\in\mathcal{V}^{m}_{\check{\alpha},r}(\check{I})$.
The construction of $q_{1}$ is straightforward (see [21]) and we have
$\left\lVert q_{1}\right\rVert_{0,\check{I}}\leq
C(\beta^{+},\beta^{-})\left\lVert u\right\rVert_{1,\check{I}}$. Now, the
second term $q_{2}$ satisfies

$q_{2}(0)=q_{2}(1)=0,\qquad\left(q_{2},v_{h}\right)_{\check{w},\check{I}}=\left(\check{u}-q_{1},v_{h}\right)_{\check{w},\check{I}},\
\forall v_{h}\in\mathcal{V}^{m-2}_{\check{\alpha},\tau^{2}(r)}(\check{I}),$

where $\check{w}=\mathcal{M}w$. Following the proof of Lemma 13, we can choose
$v_{h}=q_{2}^{\prime\prime}$ and integrate by parts to get

$-\left\lVert
wq_{2}^{\prime}\right\rVert_{0,\check{I}}^{2}=\left(\check{u}-q_{1},q_{2}^{\prime\prime}\right)_{\check{w},\check{I}}.$

We take the absolute value of each side and apply Cauchy-Schwarz inequality

$\left\lVert q_{2}^{\prime}\right\rVert_{0,\check{I}}^{2}\leq
C(\beta^{+},\beta^{-},m)\left\lVert\tilde{u}-q_{1}\right\rVert_{0,\check{I}}\left\lVert
q_{2}^{\prime\prime}\right\rVert_{0,\check{I}}.$

The inverse inequality in Lemma 3 implies that $\left\lVert
q_{2}^{\prime\prime}\right\rVert_{0,\check{I}}\leq
C(\beta^{+},\beta^{-},m)\left\lVert q_{2}^{\prime}\right\rVert_{0,\check{I}}$.
Hence,

$\left\lVert q_{2}^{\prime}\right\rVert_{0,\check{I}}\leq
C(\beta^{+},\beta^{-},m)\left\lVert\check{u}-q_{1}\right\rVert_{0,\check{I}}\leq
C(\beta^{+},\beta^{-},m)\left\lVert\check{u}\right\rVert_{1,\check{I}}.$
(56cj)

Since $q_{2}(0)=q_{2}(1)=0$, we can apply Poincaré’s inequality to obtain
$\left\lVert q_{2}\right\rVert_{0,\check{I}}<C\left\lVert
q_{2}^{\prime}\right\rVert_{0,\check{I}}$. Finally, we have

$\left\lVert\tilde{u}\right\rVert_{0,\check{I}}\leq\left\lVert
q_{1}\right\rVert_{0,\check{I}}+\left\lVert
q_{2}\right\rVert_{0,\check{I}}\leq
C(m,\beta^{+},\beta^{-})\left\lVert\check{u}\right\rVert_{1,\check{I}}.$

∎

Then, we can use Theorem 6 to derive an error bound for the Lobatto projection
$\mathscr{L}^{m}_{\alpha,r}u$ in the following theorem which confirms the
optimal approximation capability of the IFE space established in [5] by a more
complex analysis.

######  Theorem 10.

There is $C(\beta^{+},\beta^{-},m)>0$ such that the following estimate holds
for every ${u}\in\mathcal{H}^{m+1}_{\alpha,r}(I_{k_{0}})$

$|u-\mathscr{L}^{m}_{\alpha,r}u|_{i,I_{k_{0}}}\leq
C(\beta^{+},\beta^{-},m)h^{m+1-i}|u|_{m+1,I_{k_{0}}},\quad\forall
i=0,1,\dots,m.$

###### Proof.

This follows immediately from Lemma 14 and Theorem 6. ∎

###  6.2 Euler-Bernoulli Beam interface problem

In this subsection, we apply the immersed scaling argument to reanalyze the
cubic IFE space developed in [26] and [35] for solving the following interface
problem of the Euler-Bernoulli beam equation:

$\begin{cases}\beta(x)u^{(4)}(x)=f(x),\ x\in(a,\alpha)\cup(\alpha,b)\\\
u(a)=u(b)=0,\\\
u^{\prime}(a)=u^{\prime}(b)=0\end{cases}\quad\beta(x)=\begin{cases}\beta^{-}>0,&x\in(a,\alpha),\\\
\beta^{+}>0,&x\in(\alpha,b),\end{cases}$ (56ck)

where the solution $u$ satisfies the following jump conditions at $\alpha$

$[u]_{\alpha}=[u^{\prime}]_{\alpha}=[\beta u^{\prime\prime}]_{\alpha}=[\beta
u^{\prime\prime\prime}]_{\alpha}=0.$

First, let
$r=\left(1,1,\frac{\beta^{-}}{\beta^{+}},\frac{\beta^{-}}{\beta^{+}}\right)$
be fixed throughout this subsection. Then, the usual weak form of (56ck)
suggests to consider the following IFE method:

$\text{find }u_{h}\in Q^{3}_{\alpha,r}(I)\ \text{ such that }\ \left(\beta
u_{h}^{\prime\prime},v_{h}^{\prime\prime}\right)_{I}=(f,v_{h})_{I},\ \forall
v_{h}\in Q^{3}_{\alpha,r}(\mathcal{T}_{h}),$ (56cl)

where $Q^{3}_{\alpha,r}(\mathcal{T}_{h})=H^{2}_{0}(I)\cap
W^{3}_{\alpha,r}(\mathcal{T}_{h})$. We note that the IFE space
$Q^{3}_{\alpha,r}(\mathcal{T}_{h})$ as well as the method described by (56cl)
were discussed in [26] and [35], and an error analysis based on a multipoint
Taylor expansion was carried out to establish the optimality of this IFE
method in [24]. As another demonstration of the versatility of the immerse
scaling argument, we now present an alternative analysis for the optimal
approximation capability of this IFE space. This new analysis based on the
framework developed in Section 3 and Section 4 is shorter and cleaner than the
one in the literature.

As usual, for the discussion of the approximation capability of the IFE space,
we consider the interpolation on the reference element $\check{I}$ and map it
to the physical element $I_{k_{0}}$. To define the interpolation, we let
$\\{\sigma_{i}\\}_{i=1}^{4}$ be the Hermite degrees of freedom, that is,

$\displaystyle\sigma_{0}(v)=v(0),\quad\sigma_{1}(v)=v(1),\quad\sigma_{2}(v)=v^{\prime}(0),\quad\sigma_{3}(v)=v^{\prime}(1),\qquad\forall
v\in H^{2}(\check{I}).$

It is known [26, 35] that there is a basis
$\\{L^{i}_{\check{\alpha},r}\\}_{i=0}^{3}$ of
$\mathcal{V}^{3}_{\check{\alpha},r}(\check{I})$ that satisfies

$\sigma_{i}(L^{j}_{\check{\alpha},r})=\delta_{i,j},\qquad i,j=0,1,2,3.$ (56cm)

These basis functions can then be used to define an immersed Hermite
projection/interpolation operator
$\check{\mathcal{S}}_{\check{\alpha},r}:\mathcal{H}^{4}_{\check{\alpha},r}(\check{I})\rightarrow\mathcal{V}^{3}_{\check{\alpha},r}(\check{I})$
such that $\check{u}_{H}=\check{\mathcal{S}}_{\check{\alpha},r}\check{u}$ and

$\check{u}_{H}=\sum_{i=0}^{3}\sigma_{i}(\check{u})L^{i}_{\check{\alpha},r}.$
(56cn)

######  Lemma 15.

Let $\beta^{\pm}>0$ and $\check{\alpha}\in(0,1)$, then

$\displaystyle-1<L^{i}_{\check{\alpha},r}(x)<1,\qquad\forall\ x\in[0,1],\quad
i=0,1,2,3.$ (56co)

###### Proof.

See Appendix B ∎

Now, we are ready to establish that
$\\{\check{\mathcal{S}}_{\check{\alpha},r}\\}_{0<\check{\alpha}<1}$ is a
collection of uniformly bounded of RIFE projections.

######  Lemma 16.

Let $\beta^{\pm}>0,\check{\alpha}\in(0,1)$. Then there is a constant $C$
independent of $\check{\alpha}$ such that the following estimate holds for
every $\check{u}\in\mathcal{H}^{4}_{\check{\alpha},r}(\check{I})$

$\left\lVert\check{\mathcal{S}}_{\check{\alpha},r}\check{u}\right\rVert_{0,\check{I}}\leq
C\left\lVert\check{u}\right\rVert_{2,\check{I}}.$

###### Proof.

We know that $\sigma_{i}(\check{u})\leq
C\left\lVert\check{u}\right\rVert_{2,\check{I}}$ since $\check{u}\in
H^{2}(\check{I})$. Now, we apply the triangle inequality to (56cn) and Lemma
15 to get

$\left\lVert\check{\mathcal{S}}_{\check{\alpha},r}\check{u}\right\rVert_{0,\check{I}}\leq
C\left\lVert\check{u}\right\rVert_{2,\check{I}}\left(\sum_{i=0}^{3}\left\lVert
L^{i}_{\check{\alpha},r}\right\rVert_{0,\check{I}}\right)\leq
4C\left\lVert\check{u}\right\rVert_{2,\check{I}}.$

∎

Now, let
$\mathcal{S}_{\alpha,r}=\mathcal{M}^{-1}\circ\mathcal{S}_{\check{\alpha},r}\circ\mathcal{M}$
where $\mathcal{M}$ is defined in (32). By the commutative diagram in (34),
$\mathcal{S}_{\alpha,r}$ is the local immersed Hermite interpolation. Then, by
Lemma 16,
$\\{\mathcal{S}_{\alpha,r}\\}_{\check{\alpha}\in\overset{\circ}{I}_{k_{0}}}$
is a collection of uniformly bounded LIFE projections. Hence, the following
theorem follows from Theorem 6.

######  Theorem 11.

Let $\beta^{\pm}>0$, $i\in\\{0,1,2,3\\}$, $\alpha\in I_{k_{0}}$. Then, there
is a constant $C$ independent of $\alpha$ such that the following estimate
holds for every $u\in\mathcal{H}^{4}_{\alpha,r}(I_{k_{0}})$

$\left\lVert u-{\mathcal{S}}_{\alpha,r}u\right\rVert_{i,I_{k_{0}}}\leq
Ch^{3-i}|u|_{4,I_{k_{0}}}.$

This theorem establishes the optimal approximation capability of the IFE space
$Q^{3}_{\alpha,r}(\mathcal{T}_{h})$ which was first derived in [24] with a
lengthy and complex procedure.

##  7 Conclusion

In this manuscript we developed a framework for analyzing the approximation
properties of one-dimensional IFE spaces using the scaling argument. We have
applied this IFE scaling argument to establish the optimal convergence of IFE
spaces constructed for solving the acoustic interface problem, the elliptic
interface problem and the Euler-Bernoulli beam interface problem,
respectively. We are currently working on extending these results to IFE
spaces and methods for solving interface problems in two and three dimensions.

##  Appendix A Proof of Lemma 4

Our goal is to show that the ratio
$\frac{\sqrt{\check{\varphi}(0)^{2}+\check{\varphi}(1)^{2}}}{\left\lVert\check{\varphi}\right\rVert_{0,\check{I}}}$
is bounded from below by a constant $c(m,r,w)$ independent of
$\check{\alpha}$. For simplicity, let $q_{i}\in\mathcal{P}^{m}([0,1])$ be the
monomial basis $q_{i}(x)=x^{i}$ for $0\leq i\leq m$. Using the equivalence of
norms, one can show that there is $c_{1}(m)>0$ such that

$\min\left(|p(0)|,|p(1)|\right)+\sum_{i=0}^{m-1}\left|(p,q_{i})_{[0,1]}\right|\geq
c_{1}(m)\left\lVert p\right\rVert_{0,[0,1]},\qquad\forall\ p\
\in\mathcal{P}^{m}([0,1]).$ (56cp)

Unfortunately, if we extend (56cp) to $\mathcal{V}^{m}_{\check{\alpha},r}$,
then the constant on the right might depend on $\check{\alpha}$ and might grow
unboundedly as $\check{\alpha}\to 0^{+}$ or as $\check{\alpha}\to 1^{-}$. To
circumvent this issue, we will use a scaling trick similar to the one used in
the proof of Lemma 2. First, we bound
$(\check{\varphi},q_{i})_{w_{s},\check{I}^{s}}$ as shown in the following
lemma

######  Lemma 17.

Let $\tilde{m}\geq m\geq 0$, $\\{r_{k}\\}_{k=0}^{m}\subset\mathbb{R}_{+}$ and
$\check{\alpha}\in(0,1)$, there is $C(m,r,w)>0$ such that if
$\check{h}_{s}>\check{h}_{s^{\prime}}$, then

$\left|(\check{\varphi}_{s},q_{i})_{\check{I}^{s}}\right|=\left|\int_{\check{I}^{s}}\check{\varphi}_{s}(x)x^{i}dx\right|\leq
C(m,r,w)h_{s^{\prime}}\left\lVert\check{\varphi}_{s}\right\rVert_{0,\check{I}^{s}},\quad
i=0,1,\dots,m-1,\
\forall\check{\varphi}\in\mathcal{Q}^{m}_{\check{\alpha},w,r}(\check{I}).$

###### Proof.

Since $\check{\varphi}\in\mathcal{Q}^{m}_{\check{\alpha},w,r}(\check{I})$, we
have

$0=w_{s}(\check{\varphi}_{s},q_{i})_{\check{I}^{s}}+w_{s^{\prime}}(\check{\varphi}_{s^{\prime}},\mathcal{E}^{m,s^{\prime}}_{\check{\alpha},r}(q_{i}))_{\check{I}^{s^{\prime}}}.$

Then, by Cauchy-Schwarz inequality and (9), we have

$\displaystyle\left|(\check{\varphi},q_{i})_{\check{I}^{s}}\right|$
$\displaystyle=\frac{w_{s^{\prime}}}{w_{s}}\left|(\check{\varphi},\mathcal{E}^{m,s^{\prime}}_{\check{\alpha},r}(q_{i}))_{\check{I}^{s^{\prime}}}\right|\leq
C(w)\left\lVert\check{\varphi}\right\rVert_{0,\check{I}^{s^{\prime}}}\left\lVert\mathcal{E}^{m,s^{\prime}}_{\check{\alpha},r}(q_{i})\right\rVert_{0,\check{I}^{s^{\prime}}},$
$\displaystyle\leq
C(m,r,w)\sqrt{h_{s^{\prime}}}\left\lVert\check{\varphi}\right\rVert_{0,\check{I}^{s}}\sqrt{h_{s^{\prime}}}\left\lVert
q_{i}\right\rVert_{0,\check{I}^{s}},$ $\displaystyle\leq
C(m,r,w)h_{s^{\prime}}\left\lVert\check{\varphi}\right\rVert_{\check{I}^{s}}.$

∎

The previous lemma shows that $(\check{\varphi},q_{i})_{\check{I}^{s}}$ will
approach $0$ if $h_{s}$ approaches $1$. This will allow us to obtain a
restricted version of (56cp).

######  Lemma 18.

There is $\delta(m,r,w)\in(0,\frac{1}{2})$ and $C(m,r)>0$ such that if
$\min(h_{-},h_{+})<\delta(m,r,w)$, then

$|\check{\varphi}(0)|+|\check{\varphi}(1)|\geq
C(m,r)\left\lVert\check{\varphi}\right\rVert_{0,\check{I}}.\qquad\forall\check{\varphi}\in\mathcal{Q}^{m}_{\check{\alpha},w,r}(\check{I}).$

###### Proof.

We will only discuss the case where $h_{-}>h_{+}$, the other case can be
proved similarly. We define $\hat{\varphi}_{-}\in\mathcal{P}^{m}([0,1])$ as
$\hat{\varphi}_{-}(\xi)=\check{\varphi}_{-}(\check{h}_{-}\xi)$, then by the
fact that $h_{-}\geq 1/2$,

$\displaystyle|\check{\varphi}(0)|+\sum_{i=0}^{m-1}\left|\int_{0}^{h_{-}}\check{\varphi}(x)x^{i}\
dx\right|$
$\displaystyle=|\hat{\varphi}_{-}(0)|+\sum_{i=0}^{m-1}h_{-}^{i+1}\left|\int_{0}^{1}\hat{\varphi}_{-}(\xi)\xi^{i}\
d\xi\right|$
$\displaystyle\geq|\hat{\varphi}_{-}(0)|+\sum_{i=0}^{m-1}2^{-m-1}\left|(\hat{\varphi}_{-},q_{i})_{[0,1]}\right|\geq
C(m)\left(|\hat{\varphi}_{-}(0)|+\sum_{i=0}^{m-1}\left|(\hat{\varphi}_{-},q_{i})_{[0,1]}\right|\right).$

Then, by (56cp), $h_{-}\leq 1$, and (9), we have

$\displaystyle|\check{\varphi}(0)|+\sum_{i=0}^{m-1}\left|\int_{0}^{h_{-}}\check{\varphi}(x)x^{i}\
dx\right|$ $\displaystyle\geq
C(m)\left\lVert\hat{\varphi}_{-}\right\rVert_{0,[0,1]}=C(m)h_{-}^{-1/2}\left\lVert\check{\varphi}\right\rVert_{0,\check{I}^{-}}\geq
C(m)\left\lVert\check{\varphi}\right\rVert_{0,\check{I}^{-}}\geq
C_{0}(m,r)\left\lVert\check{\varphi}\right\rVert_{0,\check{I}}.$

Now, we use Lemma 17 to estimate the inner product on the left hand side:

$\sum_{i=0}^{m-1}\left|\int_{0}^{h_{-}}\check{\varphi}(x)x^{i}\ dx\right|\leq
C_{1}(m,r,w)h_{+}\left\lVert\check{\varphi}\right\rVert_{0,\check{I}^{-}}.$

We combine it with the previous inequality to get

$|\check{\varphi}(0)|\geq\left\lVert\check{\varphi}\right\rVert_{0,\check{I}}\left(C_{0}(m,r)-C_{1}(m,r,w)h_{+}\right).$

Hence, if $h_{+}\leq\delta=\min(1,\frac{C_{0}(m,r)}{2C_{1}(m,r,w)})$, then

$|\check{\varphi}(0)|\geq\frac{1}{2}C_{0}(m,r)\left\lVert\check{\varphi}\right\rVert_{0,\check{I}^{-}}$
(56cq)

A similar argument can be used to show that if $h_{+}\leq\tilde{\delta}$
(where $\tilde{\delta}$ could be different than the previous $\delta$), then

$|\check{\varphi}(1)|\geq\frac{1}{2}\tilde{C}_{0}(m,r)\left\lVert\check{\varphi}\right\rVert_{0,\check{I}^{+}}.$
(56cr)

∎

So far, we have shown that if one of the sub-elements $\check{I}^{\pm}$ is
small enough, then Lemma 4 holds. It remains to show that the lemma holds for
$\check{\alpha}\in[\delta,1-\delta]$, for which, we consider the following
sequence $\\{\mathcal{O}^{i}_{\check{\alpha},w,r}\\}_{i=0}^{m}$ by the Gram-
Schmidt process:

$\mathcal{O}^{0}_{\check{\alpha},w,r}=\mathcal{N}^{0}_{\check{\alpha},r},\quad\mathcal{O}^{i}_{\check{\alpha},w,r}=\mathcal{N}^{i}_{\check{\alpha},r}-\sum_{j=0}^{i-1}\frac{(\mathcal{N}^{i}_{\check{\alpha},r},\mathcal{O}^{j}_{\check{\alpha},w,r})_{w,\check{I}}}{(\mathcal{O}^{j}_{\check{\alpha},w,r},\mathcal{O}^{j}_{\check{\alpha},w,r})_{w,\check{I}}}\mathcal{O}^{j}_{\check{\alpha},w,r},\quad
i=1,2,\dots,m.$ (56cs)

Clearly, we have
$\mathcal{O}^{m}_{\check{\alpha},w,r}\in\mathcal{Q}^{m}_{\check{\alpha},w,r}(\check{I})$.
The following lemma shows that when $\mathcal{O}^{m}_{\check{\alpha},w,r}$ is
expressed in terms of the canonical basis
$\\{\mathcal{N}^{i}_{\check{\alpha},r}\\}_{i=0}^{m}$, the coefficients of the
expansion are rational functions.

######  Lemma 19.

Let $\tilde{m}\geq m\geq 0$, $\\{r_{k}\\}_{k=0}^{m}\subset\mathbb{R}_{+}$ and
$\check{\alpha}\in(0,1)$, the orthogonal RIFE function
$\mathcal{O}^{m}_{\check{\alpha},w,r}$ defined in (56cs) satisfies

$\mathcal{O}^{m}_{\check{\alpha},w,r}=\sum_{i=0}^{m}R_{w,r}^{i,m}(\check{\alpha})\mathcal{N}^{i}_{\check{\alpha},r}$
(56ct)

for some rational functions $\left\\{R_{w,r}^{i,m}\right\\}_{i=0}^{m}$ of
$\check{\alpha}$.

###### Proof.

We will prove Lemma 19 via strong induction. First, the case $m=0$ is obvious.
Now,we assume that For every $i=0,1,\dots,m-1$, there are rational functions
$R^{j,i}_{w,r}$ of $\check{\alpha}$ such that

$\mathcal{O}^{i}_{\check{\alpha},w,r}=\sum_{j=0}^{i}R_{w,r}^{j,i}(\check{\alpha})\mathcal{N}^{j}_{\check{\alpha},r}.$
(56cu)

To show that $\mathcal{O}^{m}_{\check{\alpha},w,r}$ satisfies (56ct), we use
the fact that
$(\mathcal{N}^{i}_{\check{\alpha},r},\mathcal{N}^{j}_{\check{\alpha},r})_{w,\check{I}}$
is a polynomial in $\check{\alpha}$. Therefore,

$\frac{(\mathcal{N}^{i}_{\check{\alpha},r},\mathcal{O}^{j}_{\check{\alpha},w,r})_{w,\check{I}}}{(\mathcal{O}^{j}_{\check{\alpha},w,r},\mathcal{O}^{j}_{\check{\alpha},w,r})_{w,\check{I}}}$
(56cv)

is a rational function of $\check{\alpha}$. Furthermore, by plugging (56cu)
into (56cs) and rearranging the terms, we get

$\mathcal{O}^{m}_{\check{\alpha},w,r}=\mathcal{N}^{m}_{\check{\alpha},r}-\sum_{j=0}^{m-1}\frac{(\mathcal{N}^{m}_{\check{\alpha},r},\mathcal{O}^{j}_{\check{\alpha},w,r})_{w,\check{I}}}{(\mathcal{O}^{j}_{\check{\alpha},w,r},\mathcal{O}^{j}_{\check{\alpha},w,r})_{w,\check{I}}}\sum_{i=0}^{j}R^{i,j}_{w,r}(\check{\alpha})\mathcal{N}^{i}_{\check{\alpha},r}=\mathcal{N}^{m}_{\check{\alpha},r}-\sum_{i=0}^{m-1}\underbrace{\left(\sum_{j=i}^{m-1}\frac{(\mathcal{N}^{m}_{\check{\alpha},r},\mathcal{O}^{j}_{\check{\alpha},w,r})_{w,\check{I}}}{(\mathcal{O}^{j}_{\check{\alpha},w,r},\mathcal{O}^{j}_{\check{\alpha},w,r})_{w,\check{I}}}R^{i,j}_{w,r}(\check{\alpha})\right)}_{:=R^{j,m}_{w,r}(\check{\alpha})}\mathcal{N}^{i}_{\check{\alpha},r}.$

From the strong induction assumption and (56cv), we conclude that
$R^{j,m}_{w,r}$ is a rational function. ∎

######  Corollary 3.

Given $m,w_{\pm}$ and $r$, the function
$\mathcal{J}_{w,r}^{m}:(0,1)\to\mathbb{R}_{+}$ defined in (56cw) is a rational
function.

Now, we are ready to prove Lemma 4. We can rewrite Lemma 18 as: there is
$\delta\in(0,\frac{1}{2})$ that depends on $m,w$ and $r$, and a constant
$C_{1}(m,r)$ such that

$\sqrt{\mathcal{O}^{m}_{\check{\alpha},w,r}(0)^{2}+\mathcal{O}^{m}_{\check{\alpha},w,r}(1)^{2}}\geq
C_{1}(m,r)\left\lVert\mathcal{O}^{m}_{\check{\alpha},w,r}\right\rVert_{0,\check{I}},\qquad\check{\alpha}\in(0,\delta)\cup(1-\delta,1).$

For $\delta\in[\delta,1-\delta]$, the following function is continuous

$\mathcal{J}^{m}_{w,r}:\check{\alpha}\mapsto\frac{\mathcal{O}^{m}_{\check{\alpha},w,r}(0)^{2}+\mathcal{O}^{m}_{\check{\alpha},w,r}(1)^{2}}{\left\lVert\mathcal{O}^{m}_{\check{\alpha},w,r}\right\rVert_{0,\check{I}}^{2}}$
(56cw)

because both of its numerator and denominator are rational functions of
$\check{\alpha}$ and the denominator is not zero. Therefore, there is
$C_{2}(m,w,r)>0$ such that

$\sqrt{\mathcal{O}^{m}_{\check{\alpha},w,r}(0)^{2}+\mathcal{O}^{m}_{\check{\alpha},w,r}(1)^{2}}\geq
C_{2}(m,w,r)\left\lVert\mathcal{O}^{m}_{\check{\alpha},w,r}\right\rVert_{0,\check{I}},\qquad\check{\alpha}\in[\delta,1-\delta].$

By letting $C(m,w,r)=\min(C_{1}(m,r),C_{2}(m,w,r))$, we know that
$\mathcal{O}^{m}_{\check{\alpha},w,r}$ satisfies inequality (24) stated in
Lemma 4. Consequently, the estimates in (24) of Lemma 4 is true for every
function in $\mathcal{Q}^{m}_{\check{\alpha},w,r}(\check{I})$ because it is a
one-dimensional space, and Lemma 4 is proven.

##  Appendix B Proof of Lemma 15

Let us start with $p=L^{0}_{\check{\alpha},r}$, we have

$p(0)=1,\quad p(1)=0,\quad p^{\prime}(0)=0,\quad p^{\prime}(1)=0.$

By Theorem 1, $p^{\prime}$ does not change sign in $(0,1)$ since
$p^{\prime}\in\mathcal{V}^{2}_{\check{\alpha},\tau(r)}(\check{I})$ and
$p^{\prime}(0)=p^{\prime}(1)=0$. Therefore, $p$ is monotonically decreasing
from $p(0)=1$ to $p(1)=0$. The same argument applies to
$L^{1}_{\check{\alpha},r}$.

Next, we show that $q=L^{2}_{\check{\alpha},r}$ is bounded between $0$ and
$1$. We have

$q(0)=0,\quad q(1)=0,\quad q^{\prime}(0)=1,\quad q^{\prime}(1)=0.$

By Rolle’s theorem, there is $c\in(0,1)$ such that $q^{\prime}(c)=0$. By
Theorem 1, $c$ the only root of $q^{\prime}$ in $(0,1)$. Now, by the
generalized Rolle’s theorem, there is $d\in(c,1)$ such that
$q^{\prime\prime}(d^{-})q^{\prime\prime}(d^{+})\leq 0$. If
$d\not=\check{\alpha}$, then
$q^{\prime\prime}(d^{-})=q^{\prime\prime}(d^{+})=q^{\prime\prime}(d)$ because
$q$ is a polynomial on either sides of $\check{\alpha}$. In this case we have
$q^{\prime\prime}(d)=0$. If $d=\check{\alpha}$, then
$q^{\prime\prime}(d^{-})q^{\prime\prime}(d^{+})\leq 0$ and jump condition
implies

$\displaystyle\frac{\beta^{-}}{\beta^{+}}\big{(}q^{\prime\prime}(\check{\alpha}^{-})\big{)}^{2}\leq
0$

from which we have
$q^{\prime\prime}(\check{\alpha}^{-})=0=q^{\prime\prime}(\check{\alpha}^{+})$.
Hence, $q^{\prime\prime}(d)=0$. Furthermore, by Theorem 1, $d$ is the only
root of $q^{\prime\prime}$ since
$q^{\prime\prime}\in\mathcal{V}^{1}_{\check{\alpha},\tau^{2}(r)}(\check{I})$.
Since $q^{\prime\prime}$ is a linear polynomial on either sides of
$\check{\alpha}$, the jump condition satisfied by $q$ further implies that
$q^{\prime\prime}$ does not change its sign $(0,d)$ and $(d,1)$. Because
$q^{\prime}(0)=1$ and $q^{\prime}(c)=0$ and $0<c<d$, we know that $q^{\prime}$
is decreasing on $(0,d)$ but increasing on $(d,1)$. These further imply
$q^{\prime}(x)\in[0,1]$ for $x\in[0,c]$ and $q^{\prime}(x)\leq 0$ for
$x\in[c,d]$; hence, $q(x)\leq q(c)$ for all $x\in[0,d]$. Furthermore, since
$q^{\prime}(d)\leq 0,q^{\prime}(1)=0$ and $q^{\prime}$ is monotonic on
$[d,1]$, we know that $q^{\prime}(x)\leq 0$ for all $x\in[d,1]$. Hence,
$0=q(1)\leq q(x)\leq q(d)\leq q(c)$ for all $x\in[d,1]$. Consequently,
$q(c)\geq q(x)$ for $x\in[0,1]$. In addition, since $q$ has no local minimum
point on $(0,d)$, we have $q(x)\geq\min\\{q(0),q(d)\\}\geq 0$ for all
$x\in[0,d]$. Thus, $q(x)\geq 0~{}\forall x\in[0,1]$. On the other hand,

$\displaystyle q(x)\leq
q(c)=\int_{0}^{c}q^{\prime}(x)dx\leq\int_{0}^{c}1dx=c<1~{}~{}\forall
x\in[0,1],$

The last two estimates lead us to conclude that $0\leq
q(x)=L^{2}_{\check{\alpha},r}(x)<1$. As for $L^{3}_{\check{\alpha},r}$, we
note that

$\displaystyle
L^{3}_{\check{\alpha},r}(x)=-L^{2}_{1-\check{\alpha},\tilde{r}}(1-x),\qquad\text{where
}\tilde{r}=\\{r_{i}^{-1}\\}_{i=0}^{3}$

which leads to $L^{3}_{\check{\alpha},r}(x)\in[-1,0]$.

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