# Solving the unique continuation problem for Schrödinger equations with low
regularity solutions using a stabilized finite element method

Erik Burman  Department of Mathematics, University College London, UK
<EMAIL_ADDRESS>,  Mingfei Lu  Department of Mathematics, University
College London, UK<EMAIL_ADDRESS>and  Lauri Oksanen  Department of
Mathematics and Statistics, University of Helsinki,Finland
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we consider the unique continuation problem for the Schrödinger
equations. We prove a Hölder type conditional stability estimate and build up
a parameterized stabilized finite element scheme adaptive to the a priori
knowledge of the solution, achieving error estimates in interior domains with
convergence up to continuous stability. The approximability of the scheme to
solutions with only $H^{1}$-regularity is studied and the convergence rate for
solutions with regularity higher than $H^{1}$ is also shown. Comparisons in
terms of different parameterization for different regularities will be
illustrated with respect to the convergence and condition numbers of the
linear systems. Finally, numerical experiments will be given to illustrate the
theory.

###### 2020 Mathematics Subject Classification:

Primary

##  1\. Introduction

We will consider the Unique Continuation Problem for Schrödinger equations

$-\Delta u+Pu=f,$

and develop a stabilized finite element method to solve it numerically.
Similar work for Poisson’s equation can be found in [MR3134434,
burman2018solving], and that for Helmholtz equation in [burman2019unique].
These works can be seen as developments of the quasi-reversibility method
introduced in [Bour05] where numerical stability and Tikhonov regularization
have been separated in order to allow for error estimates using conditional
stability. In particular in [Bour05] the convergence for vanishing
regularization was proven assuming only that the solution is in
$H^{1}(\Omega)$ for unperturbed data. The analysis of [MR3134434,
burman2018solving, burman2019unique] typically requires stronger regularity
assumptions in order to show that the numerical method has a convergence
order. However the instability of the problem excludes the extension of their
analysis of the method to the low regularity case, even in the absence of
perturbations. The objective of the present work is to show how to design the
stabilized methods so that low regularity can be handled in a similar fashion,
reflecting the conditional stability and the approximation of the finite
element space in the regularity class of the exact solution. Observe that the
design of the method requires a priori knowledge of this regularity to obtain
the best convergence rate. Let us also point out that similar estimates
combining conditional stability and approximation have recently been obtained
in [DMS23].

The mathematical formulation of the unique continuation problem for
Schrödinger equations is as follows:
$\omega\subset\Omega\subset\mathbb{R}^{n}$ be domains. Let $P\in
L^{\infty}(\Omega)$ be the potential. Find $u\in H^{1}(\Omega)$, such that

(1.1) $\left\\{\begin{array}[]{rcc}-\Delta u+Pu&=f&\texttt{in $\Omega$}\\\
u&=q&\texttt{in $\omega$}\end{array}.\right.$

It is shown, in Chapter 2, that if the solution exists then it is unique.
However, since such unique continuation problem has been shown to be ill-
posed, in terms of continuity, we can at most expect a conditional stability
estimate that reads:

$||u||_{H^{1}(B)}\leq C(P)(||-\Delta
u+Pu||_{L^{2}(\Omega)}+||u||_{L^{2}(\omega)})^{\kappa}(||u||_{H^{1}(\Omega)})^{1-\kappa},$

for $B\setminus\omega\subset\subset\Omega$. When $B\setminus\omega$ touches
the boundary, the best estimate we could achieve is a logarithmic estimate,
see [john1960continuous]. In this paper, we will focus on the former case,
i.e. the area of interest $\overline{B\setminus\omega}\subset\Omega$.

Our aim is to build a stabilized finite element method to solve (1.1)
computationally. Observe that with the least a priori knowledge $u\in
H^{1}(\Omega)$ we will only be able to show that the scheme converges weakly
in both $L^{2}$ and $H^{1}$ norms without a convergence rate. However such low
regularity can be a result from three different reasons:

1\. low regularity of the right hand side $f$;

2\. low regularity of the geometry $\Omega$;

3\. low regularity of the unknown boundary data.

Let us first assume that the unknown boundary data are smooth and that
$\Omega$ is a convex polygonal. If only the right hand side has low
regularity, $f\in H^{-1}(\Omega)$ we can decompose the solution into a
singular part $u_{s}\in H^{1}_{0}(\Omega)$, solution to

(1.2) $-\Delta u_{s}+Pu_{s}=f\texttt{ in $\Omega$},$

and $u_{r}\in H^{r}(\Omega)$ with $r\geq 2$ such that $u_{r}=q-u_{s}$ in
$\omega$, and

(1.3) $-\Delta u_{r}+Pu_{r}=0\texttt{ in $\Omega$},$

Clearly in this situation we can solve (1.2) using the standard finite element
method with piece-wise affine elements, leading to the error estimate

(1.4) $||u_{s}-u_{h,s}||_{\Omega}\leq Ch||f||_{H^{-1}(\Omega)}.$

Since $u_{r}$ is smooth we can apply the primal-dual stabilised method (see
Chapter 3), with elements of polynomial degree less than or equal to $p$, to
obtain the optimal approximation $u_{h,r}$ satisfying

$||u_{r}-u_{h,r}||_{B}\leq C(P)h^{\kappa t}||u_{r}||_{H^{t+1}(\Omega)},$

where $t=\min\\{p,r-1\\}$, $B\subset\subset\Omega$ and $\kappa\in(0,1)$ a
coefficient depending on the distance from the boundary of $B$ to the boundary
of $\Omega$. As this distance goes to zero, so does $\kappa$.

It follows that in the special case where the only cause of poor regularity is
the right hand side $f$, we can still obtain an error estimate for
$u_{h}=u_{h,s}+u_{h,r}$,

$||u-u_{h}||_{B}\lesssim h||f||_{H^{-1}(\Omega)}+h^{\kappa
t}||u_{r}||_{H^{t+1}(\Omega)}.$

Hence for unperturbed data it is possible to achieve up to first order
convergence in spite of the singular right hand side. Of course these
arguments do not work for the second and third points above. If $\Omega$ is
not smooth enough then the estimate (1.4) fails due to lack of stability of
the adjoint equation and if the boundary data has insufficient smoothness then
$u_{r}\in H^{r}(\Omega)$ can not hold with $r\geq 2$. The objective of the
present work is to propose an error analysis that is valid for all sources of
low regularity. We will follow a standard optimization approach by optimizing
the stabilized Lagrange functional $L_{h}(u_{h},z_{h})$ at the discrete level,
namely,

$\displaystyle L_{h}(u_{h},z_{h}):=$
$\displaystyle\frac{1}{2}||u_{h}-q||^{2}_{\omega}+\textit{Primal
stabilizer}-\textit{Dual stabilizer}$
$\displaystyle+a(u_{h},z_{h})-<f,z_{h}>_{\Omega}.$

The stabilization method combines tools known from finite element methods to
ensure a discrete inf-sup condition and Tikhonov type regularization, see, for
example [engl1996regularization, Chapter 5]. Similar type of approaches have
been used in [burman2019unique, burman2020stabilized]. The main differences
between this paper and [burman2019unique, burman2020stabilized] are that we
build up a parameterized approach adaptive to the prior knowledge for the
smoothness of the exact solution $u$, for $u\in H^{1}(\Omega)$ or $u\in
H^{s}(\Omega)$, with $s>1$. The main results are that for $u\in
H^{1}(\Omega)$, we prove our scheme converges weakly, with error estimates on
residual quantities only,

$||u_{h}-q||_{L^{2}(\omega)}\leq Ch^{\alpha}||u||_{H^{1}(\Omega)},$

and

$||\mathcal{L}u_{h}-f||_{H^{-2}(\Omega)}\leq
C(h+h^{\frac{\tau}{2}})||u||_{H^{1}(\Omega)},$

where $0<\alpha\leq 1,0<\tau\leq 2$ are parameters we will use in the scheme.
Moreover, for $u\in H^{s}(\Omega)$ with $s>1$ ($s$ may be fractional), we will
show that our scheme has a convergence matching the approximation order of the
finite element space and the conditional stability estimate, namely,

$||u-u_{h}||_{H^{1}(B)}\leq C(P)h^{\kappa(s-1)}||u||_{H^{s}(\Omega)}.$

##  2\. Continuous stability estimate

Our stabilized finite element method relies on several conditional stability
estimates at the continuous level. The layout of needed estimates follows
[burman2019unique] for Helmholtz equations. We will state similar estimates
for Schrödinger equations that we are interested in. These estimates are based
on the following well known Carleman estimate:

######  Lemma 2.1 (Carleman estimate).

car let $\rho\in C^{3}(\Omega)$ and $K\subset\Omega$ be a compact subset and
contains no critical point of $\rho$. Let $\alpha>0$ and
$\phi=e^{\alpha\rho}$. Then for some large $\alpha$ there exists $\tau_{0}\geq
1$ and $C>0$ depending on $\alpha$ and $\rho$, such that for all
$\tau\geq\tau_{0}$ and $w\in C^{2}_{0}(K)$,

$\int_{K}e^{2\tau\phi}(\tau|\nabla w|^{2}+\tau^{3}|w|^{2})dx\leq
C\int_{K}e^{2\tau\phi}|\Delta w|^{2}dx.$

###### Proof.

See [nechita2020unique, Corollary 2.3].

∎

Now we give a lemma optimizing exponential parameters and use it to derive
three Hölder type continuous stability estimates with optimized parameters,
the first one is the estimate for $H^{1}$-norm in the $H^{2}$ space, the
second one is the estimate for $H^{1}$-norm in the $H^{1}$-space, and the last
one is for $L^{2}$-norm in the $H^{1}$ space. The latter two estimates can be
used in showing convergence of the stabilized finite element method, and their
proof will be based on the first estimate.

######  Lemma 2.2.

Suppose that $A$, $B$, $D$, $c$ be positive numbers and $\kappa\in(0,1)$
satisfying $D\leq B$ and $D\leq e^{c(1-\kappa)\lambda}A+e^{-c\kappa\lambda}B$
for all $\lambda\geq\lambda_{0}>0$. Then for any $\epsilon$ positive and small
there exists constant $C$ depending on $c,\epsilon$ such that

$D\leq e^{C\kappa^{1-\epsilon}(\lambda_{0}+1)}A^{\kappa}B^{1-\kappa}.$

###### Proof.

See [le2012carleman, Lemma 5.2].

∎

Using Lemma 2.2, we will be able to prove the following estimate.

######  Corollary 2.3 (Three-ball inequality).

tbi Let $x_{0}\in\Omega$ and $0<r_{1}<r_{2}<d(x_{0},\partial\Omega)$. Define
$B_{j}=B(x_{0},r_{j})$, $j=1,2$. Then there exists $C>0$ and $\kappa\in(0,1)$
depending only on $\Omega$, such that for $u\in H^{2}(\Omega)$ and any small
$\epsilon$ it holds that:

$||u||_{H^{1}(B_{2})}\leq\exp(C\kappa^{1-\epsilon}(||P||^{\frac{2}{3}}_{L^{\infty}(\Omega)}+1))(||u||_{H^{1}(B_{1})}+||-\Delta
u+Pu||_{L^{2}(\Omega)})^{\kappa}||u||_{H^{1}(\Omega)}^{1-\kappa}.$

###### Proof.

See Appendix A ∎

######  Remark 2.4.

Note that for simplicity, we only proved in tbi the estimate holds for this
specific geometry, say, $B_{1}\subset B_{2}\subset\Omega$, but actually it can
hold for a larger class of geometric settings. Let such a geometry be
$\omega\subset B\subset\Omega$.

######  Corollary 2.5 (Unique continuation property).

Assume problem (1.1) has a solution, and the conditional stability estimate in
tbi holds for any $B$ such that $\omega\subset B\subset\Omega$. Then the
solution is unique.

###### Proof.

Let $u$, $v$ be two different solutions to (1.1), consider the difference
$w=u-v$, we have

$\left\\{\begin{array}[]{rcc}-\Delta w+Pw&=0&\texttt{in $\Omega$}\\\
u&=0&\texttt{in $\omega$}\end{array}.\right.$

By the estimate in tbi, the right hand side vanishes, and thus the conclusion
follows.

∎

Note also that in tbi, $u$ is required to be $H^{2}(\Omega)$ because
$||-\Delta u+Pu||_{L^{2}(\Omega)}$ is used on the right-hand side. We want to
generalize the estimate to make it hold for all $H^{1}(\Omega)$ functions.

######  Corollary 2.6.

co2 Let $B_{1}\subset B_{2}\subset\Omega$ be defined as in tbi. Let $u\in
H^{1}(\Omega)$. Then the following estimate holds

$\displaystyle||u||_{H^{1}(B_{2})}\leq$ $\displaystyle
C(P)(||u||_{L^{2}(B_{1})}+||-\Delta u+Pu||_{H^{-1}(\Omega)})^{\kappa}$
$\displaystyle\cdot(||u||_{L^{2}(\Omega)}+||-\Delta
u+Pu||_{H^{-1}(\Omega)})^{1-\kappa}$

Furthermore, if $P\geq 0$ on $\Omega$, then $C(P)$ is the same as that in tbi.

###### Proof.

See Appendix B. ∎

In co2 we derive a conditional stability estimate for $u\in H^{1}(\Omega)$.
Now we give another bound for $||u||_{L^{2}(B_{2})}$, $\forall u\in
H^{1}(\Omega)$.

######  Corollary 2.7.

L2error Let the geometric setting be the same as in tbi. Then there exists
$C>0$ and $\kappa\in(0,1)$ depending only on $\Omega$, such that for $u\in
H^{1}(\Omega)$ and any small $\epsilon$ it holds that:

$||u||_{L^{2}(B_{2})}\leq C(P)\left(||u||_{L^{2}(B_{1})}+||-\Delta
u+Pu||_{H^{-1}(\Omega)}\right)^{\kappa}||u||_{L^{2}(\Omega)}^{1-\kappa},$

where
$C(P)=\exp\left(C\kappa^{1-\epsilon}(||P||^{\frac{2}{3}}_{L^{\infty}(\Omega)}+1)\right)$.

###### Proof.

See Appendix C. ∎

Note that by shifting $H^{1}(\Omega)$ norm to $L^{2}(\Omega)$ norm, we get an
exact constant depending on $P$ but win nothing in the order of $\kappa$. This
means the $L^{2}$-norm error in our finite element method generally doesn’t
give a higher order of convergence than the $H^{1}$-norm error.

##  3\. Stabilized finite element method

In this section we propose a stabilized $H^{1}$-conforming finite element
method for problem (1.1), with priori knowledge that $u\in H^{s}(\Omega)$,
$\forall s\geq 1$. We will first form an equivalent optimization problem, then
stabilize the optimization problem at the discrete level. Then we show that
the method gives an convergence rates up to the conditional stability. We will
also discuss the stability of the discrete system and give precision
thresholds.

Lagrange optimization problem. In this part we derive the equivalent
optimization problem to problem (1.1) and give some useful propositions we
will use repeatedly. From now on, we will make some simplifications on
notations. Let $(\cdot,\cdot)_{U},\langle\cdot,\cdot\rangle_{U}$ denote the
$L^{2}$ and $H^{1}$ inner product on some domain $U$, respectively. Let
$\mathcal{L}$ be the elliptic operator $-\Delta+P$, and $<\cdot,\cdot>_{U}$ be
the $H^{-1}(U)-H_{0}^{1}(U)$ pair.

######  Proposition 3.1.

Let problem (1.1) have a unique solution, then $u\in H^{1}(\Omega)$ solves
problem (1.1) iff $u$ solves

(3.1) $\begin{array}[]{cccc}\underset{u\in
H^{1}(\Omega)}{\min}\frac{1}{2}||u-q||^{2}_{\omega}&\textit{subject
to}&a(u,w)=<f,w>_{\Omega},&\forall w\in H^{1}_{0}(\Omega)\end{array}$

Now we construct the Lagrange functional for the optimization problem (3.1)
with the PDE constraint. Let

$L(u,z):=\frac{1}{2}||u-q||^{2}_{\omega}+a(u,z)-<f,z>_{\Omega},\;\;\;(u,z)\in
H^{1}(\Omega)\times H^{1}_{0}(\Omega),$

where $z\in H^{1}_{0}(\Omega)$ acts as a Lagrange multiplier. The
corresponding Euler-Lagrange equation reads:

(3.2) $\left\\{\begin{array}[]{l}0=\partial_{u}Lv=(u-q)_{\omega}+a(v,z)\\\
0=\partial_{z}Lw=a(u,w)-<f,w>_{\Omega}\end{array}\right.\;\;\forall(v,w)\in
H^{1}(\Omega)\times H^{1}_{0}(\Omega).$

The continuous problem is clearly ill-posed because the equivalent PDE problem
is ill-posed. Instead of regularizing the problem at this stage we will
transfer the functional optimization problem to the discrete level and then
discuss the stabilization method.

Finite element settings. From now on we will consider for simplicity that
$\Omega\subset\mathbb{R}^{d}$, $d\in\\{2,3\\}$, is a convex
polygonal/polyhedral domain. Let $\\{\mathcal{T}_{h}\\}_{h>0}$ be a quasi-
uniform family of shape-regular meshes covering $\Omega$, each triangulation
$\mathcal{T}_{h}$ containing elements $K$ with maximal diameter $h$. Let
$p\geq 1$ denote the polynomial degree of approximation and let
$\mathbb{P}_{p}(K)$ be the space of polynomials of degree at most $p$ defined
on $K$. Let the $H^{1}$-conforming finite element space

$V^{p}_{h}:=\\{v_{h}\in H^{1}(\Omega):v_{h}|_{K}\in\mathbb{P}_{p}(K),\forall
K\in\mathcal{T}_{h}\\}$

and its subspace with homogeneous boundary conditions

$W_{h}^{p}:=V^{p}_{h}\cap H^{1}_{0}(\Omega).$

Moreover, define also $V_{h}^{0}:=\\{v_{h}\in
L^{2}(\Omega):v_{h}|_{K}=C_{k}\in\mathbb{R},\forall K\in\mathcal{T}_{h}\\}$ be
the space of piece-wise constant functions.

The corresponding discrete problem will be the restriction of $(u,z)$ on
$V^{p}_{h}\times W^{p}_{h}$. One can consider the Lagrange functional

$L^{singular}_{h}(u_{h},z_{h}):=\frac{1}{2}h^{-2\alpha}||u_{h}-q||^{2}_{\omega}+a(u_{h},z_{h})-<f,z_{h}>_{\Omega},\;\;\;(u_{h},z_{h})\in
V^{p}_{h}\times W^{p}_{h},$

where $h^{-2\alpha}$ is the mesh-specific weighting term between interior
difference $u_{h}-q$ and the PDE constraint. We will specify the choice of
$\alpha$ later.

Finding saddle points of the above Lagrange functional may still produce a
singular linear system. Thus we introduce primal and dual stabilizers, which
are bi-linear forms on $V^{p}_{h}$ and $W^{p}_{h}$, $s_{h}(\cdot,\cdot)$ and
$s_{h}^{*}(\cdot,\cdot)$, and a complimentary term $G(\cdot,\cdot)$ to keep
consistency. Now introduce the stabilized Lagrange functional:

$\displaystyle L_{h}(u_{h},z_{h}):=$
$\displaystyle\frac{1}{2}h^{-2\alpha}||u_{h}-q||^{2}_{\omega}+\frac{1}{2}s_{h}(u_{h},u_{h})$
$\displaystyle-\frac{1}{2}s_{h}^{*}(z_{h},z_{h})+a(u_{h},z_{h})-<f,z_{h}>_{\Omega}+G(f,u_{h}).$

Then the corresponding Euler-Lagrange equations read: for any
$(v_{h},w_{h})\in V_{h}^{p}\times W^{p}_{h}$,

$\left\\{\begin{array}[]{l}0=\partial_{u_{h}}L_{h}v_{h}=h^{-2\alpha}(u_{h}-q)_{\omega}+s_{h}(u_{h},v_{h})+a(v_{h},z_{h})+G(f,v_{h})\\\
0=\partial_{z_{h}}L_{h}w_{h}=a(u_{h},w_{h})-s_{h}^{*}(z_{h},w_{h})-<f,w_{h}>_{\Omega},\end{array}\right.$

and the stabilized finite element method reads: find $(u_{h},z_{h})\in
V^{p}_{h}\times W^{p}_{h}$, such that $\forall(v_{h},w_{h})\in V^{p}_{h}\times
W^{p}_{h}$,

(3.3)
$\left\\{\begin{array}[]{rl}h^{-2\alpha}(u_{h},v_{h})_{\omega}+s_{h}(u_{h},v_{h})+a(v_{h},z_{h})&=h^{-2\alpha}(q,v_{h})_{\omega}+G(f,v_{h})\\\
a(u_{h},w_{h})-s_{h}^{*}(z_{h},w_{h})&=<f,w_{h}>_{\Omega}\end{array}\right..$

Stabilization. Now let us introduce our choice of stabilizers. Let
$\mathcal{L}_{h}$ be the discrete Shrödinger operator, namely, $\forall
v_{h}\in V^{p}_{h}$,

$\mathcal{L}_{h}v_{h}:=\sum_{K}(-\Delta+P)v_{h}|_{K}.$

For a given mesh $\mathcal{T}_{h}$, we consider the set of all
interior(relative to $\Omega$) edges/faces of elements, denoted by
$\mathcal{F}_{i}$, and the jump function on each edge/face $F$ by
$\llbracket\cdot\rrbracket_{F}$, omitting its subscript whenever there is no
confusion. For $v_{h}\in V^{p}_{h}$, we denote the jump function of its normal
gradient across $F\in\mathcal{F}_{i}$ by

$\llbracket\nabla v_{h}\cdot n\rrbracket_{F}:=\nabla v_{h}\cdot
n_{1}|_{K_{1}}+\nabla v_{h}\cdot n_{2}|_{K_{2}},$

with $K_{1},K_{2}\in\mathcal{T}$ being two elements such that $K_{1}\cap
K_{2}=F$, and $n_{1},n_{2}$ the outward pointing normals with respect to
$K_{1}$ and $K_{2}$. Note that for $v\in H^{2}(\Omega)$, $\llbracket\nabla
v\cdot n\rrbracket_{F}=0$ is well defined. $\forall v_{h}\in V^{p}_{h}$,
define

$\mathcal{J}_{h}(u_{h},v_{h})=\sum_{F}\int_{F}h\llbracket\nabla v_{h}\cdot
n\rrbracket\cdot\llbracket\nabla v_{h}\cdot n\rrbracket_{F}dS.$

Now define $s_{h}:V^{p}_{h}\times V_{h}^{p}\to\mathbb{R}$ by

(3.4)
$s_{h}(u_{h},v_{h}):=\mathcal{J}_{h}(u_{h},v_{h})+h^{2}(\mathcal{L}_{h}u_{h},\mathcal{L}_{h}v_{h})_{\Omega}+h^{2(s-1)}\langle
u_{h},v_{h}\rangle_{\Omega},$

$s^{*}_{\eta}(z_{h},w_{h}):W^{p}_{h}\times W_{h}^{p}\to\mathbb{R}$ by

(3.5)
$s^{*}_{\eta}(z_{h},w_{h}):=h^{2\eta}\mathcal{J}_{h}(z_{h},w_{h})+h^{2\eta}\int_{\partial\Omega}h\partial_{n}z_{h}\partial_{n}w_{h}dS+h^{2\eta}(h\mathcal{L}_{h}z_{h},h\mathcal{L}_{h}w_{h}),$

and $s^{*}_{h}$ by

(3.6) $s_{h}^{*}(z_{h},w_{h}):=s^{*}_{\eta}(z_{h},w_{h})+h^{\tau}\langle
z_{h},w_{h}\rangle_{\Omega},$

where $h^{2\eta}$ and $h^{\tau}$ is the weights with respect to $s_{\eta}^{*}$
and $\langle z_{h},w_{h}\rangle_{\Omega}$ in $s^{*}_{h}$.

Now we define the consistency term $G(f,v_{h})$. Let $f_{h}$ be the projection
from $H^{-1}(\Omega)$ to $W^{p}_{h}$ such that for any $w_{h}\in W^{p}_{h}$,

$<f,w_{h}>_{\Omega}=(f_{h},w_{h})_{\Omega}.$

Specifically, if $f\in L^{2}(\Omega)$, we have

$(f_{h},f_{h})_{\Omega}=(f,f_{h})_{\Omega}\leq||f||_{\Omega}||f_{h}||_{\Omega},$

which gives

(3.7) $||f_{h}||_{\Omega}\leq||f||_{\Omega}.$

Define $G:H^{-1}(\Omega)\times V^{p}_{h}\to\mathbb{R}$ by

(3.8) $G(f,v_{h})=h^{2}(f_{h},\mathcal{L}_{h}v_{h})_{\Omega}.$

Inverse and trace inequalities. Now we give some useful inequalities and
approximation results (see, for example, [di2011mathematical]), and then
introduce our exact choice of stabilizers.

(1) Continuous trace inequality.

(3.9) $||v||_{\partial K}\leq
C(h^{-\frac{1}{2}}||v||_{K}+h^{\frac{1}{2}}||\nabla v||_{K}),\;\;\forall v\in
H^{1}(K).$

(2) Discrete trace inequality.

(3.10) $||\nabla v_{h}\cdot n||_{\partial K}\leq Ch^{-\frac{1}{2}}||\nabla
v_{h}||_{K},\;\;\forall v_{h}\in\mathbb{P}_{p}(K).$

(3) Inverse inequality.

(3.11) $||\nabla v_{h}||_{K}\leq Ch^{-1}||v_{h}||_{K},\;\;\forall
v_{h}\in\mathbb{P}_{p}(K).$

Interpolation. We now give interpolants we will use repeatedly.

(1) The Scott-Zhang operator $\pi_{sz}:H^{\frac{1}{2}}(\Omega)\to V_{h}^{p}$.
It keeps homogeneous boundary condition and has the optimal approximation
property, see [ciarlet2013analysis]:

$||v-\pi_{sz}v||_{H^{m}(\mathcal{T}_{h})}\leq Ch^{s-m}|v|_{H^{s}(\Omega)},$

whenever $v\in H^{s}(\Omega)$ with $\frac{1}{2}<s\leq p+1$ and
$m\in\\{0:[s]\\}$.

(2) $L^{2}$-projection $\pi_{0}:L^{2}\to V_{h}^{0}$. It has local optimal
approximation property, see [ern2021finite, Theorem 18.16]:

$||v-\pi_{0}v||_{K}\leq Ch^{s}|v|_{H^{s}(K)},$

whenever $K\in\mathcal{T}_{h}$, $v\in H^{s}(K)$ with $0\leq s\leq 1$. Note
that $V_{h}^{0}$ is not $H^{1}$-conforming.

(3) $H^{1}$-conforming $L^{2}$-projection $\pi_{h}:L^{2}\to V_{h}^{p}$ with
$p\geq 1$. It is the best approximation in the $L^{2}$-norm, and thus it has
optimal approximation property in $L^{2}$-norm

$||v-\pi_{h}v||_{\Omega}\leq Ch^{s}|v|_{H^{s}(\Omega)},$

whenever $v\in H^{s}(\Omega)$ with $0\leq s\leq p+1$.

Properties of the stabilized system. We first show that
$\mathcal{J}_{h}(v_{h},v_{h})$ and
$h^{2}(\mathcal{L}_{h}v_{h},\mathcal{L}_{h}v_{h})_{\Omega}-G(f_{h},v_{h})$ are
weakly consistent.

######  Proposition 3.2.

weakcons Let $\pi_{sz}$ be the Scott-Zhang operator and $v\in H^{s}(\Omega)$
for $s\geq 1$. Then there is a constant $C$ such that

$\mathcal{J}_{h}(\pi_{sz}v,\pi_{sz}v)\leq
Ch^{2(s-1)}||v||^{2}_{H^{s}(\Omega)}.$

Moreover, if $s<2$ we have

$||h\mathcal{L}_{h}(\pi_{sz}v)||_{\Omega}\leq
C(1+h||P||_{L^{\infty}})h^{s-1}||v||_{H^{s}(\Omega)},$

and if $s\geq 2$, we have

$||h\mathcal{L}_{h}(\pi_{sz}v-v)||_{\Omega}\leq
C(1+h^{2}||P||_{L^{\infty}})h^{s-1}||v||_{H^{s}(\Omega)}.$

###### Proof.

Step1. We give the estimate for the jump of gradients. Let $\pi_{h}^{n}$ be
the product of $n$ $\pi_{h}$ acting component-wise on $(L^{2}(\Omega))^{n}$
Since the range of $\pi_{h}$ is in $H^{1}(\Omega)$,
$\llbracket\pi_{h}w\rrbracket=0$ for all $w\in L^{2}(\Omega)$. This gives

$\displaystyle\mathcal{J}_{h}(\pi_{sz}v,\pi_{sz}v)$
$\displaystyle=\underset{F\in\mathcal{F}_{i}}{\sum}\int_{F}h\llbracket(\nabla\pi_{sz}v-\pi_{h}^{n}\nabla\pi_{sz}v)\cdot
n\rrbracket_{F}\llbracket(\nabla\pi_{sz}v-\pi_{h}^{n}\nabla\pi_{sz}v)\cdot
n\rrbracket_{F}dS$ $\displaystyle\leq
C\sum_{K}||\nabla\pi_{sz}v-\pi_{h}^{n}\nabla\pi_{sz}v_{h}||^{2}_{K}$
$\displaystyle\leq C(||\nabla(\pi_{sz}v-v)||^{2}_{\Omega}+||\nabla
v-\pi_{h}^{n}\nabla
v)||^{2}_{\Omega}+||\pi_{h}^{n}\nabla(v-\pi_{sz}v)||^{2}_{\Omega})$
$\displaystyle\leq Ch^{2(s-1)}||v||^{2}_{H^{s}(\Omega)}$

where the first inequality is from (3.9), (3.11) and the third inequality is
from the $H^{1}$ optimal approximation property of $\pi_{sz}$ and the $L^{2}$
optimal approximation property of $\pi^{n}_{h}$.

Step 2. Now we only need to give the estimate for
$||h\mathcal{L}_{h}(\pi_{sz}v)||_{\Omega}$ when $1\leq s<2$, since when $s\geq
2$ the estimate is given immediately by the optimal approximation property of
$\pi_{sz}$. Let $\pi_{0}^{n}$ be the product of $n$ $\pi_{0}$, and
$\nabla\cdot\pi_{0}^{n}\nabla i_{h}v=0$ since $\pi_{0}$ maps a function to
piece-wise constants. Let $\Delta_{h}$ be the element-wise Laplacian, we have

$\displaystyle||h\Delta_{h}\pi_{sz}v||_{\Omega}$
$\displaystyle=(\sum_{K}||h\nabla\cdot\nabla\pi_{sz}v||^{2}_{K})^{\frac{1}{2}}$
$\displaystyle=(\sum_{K}||h\nabla\cdot(\nabla\pi_{sz}v-\pi_{0}^{n}\nabla\pi_{sz}v)||_{K}^{2})^{\frac{1}{2}}$
$\displaystyle\leq
C(\sum_{K}||\nabla\pi_{sz}v-\pi_{0}^{n}\nabla\pi_{sz}v||^{2}_{K})^{\frac{1}{2}}$
$\displaystyle\leq C\left(\sum_{K}(||\nabla(\pi_{sz}v-v)||^{2}_{K}+||\nabla
v-\pi_{0}^{n}\nabla
v||^{2}_{K}+||\pi_{0}^{n}\nabla(v-\pi_{sz}v)||^{2}_{K})\right)^{\frac{1}{2}}$
$\displaystyle\leq Ch^{s-1}||v||_{H^{s}(\Omega)},$

where the first inequality is from (3.11), the third inequality is from
optimal approximation property of $\pi_{sz}$ and local optimal approximation
property of $\pi_{0}$, and the last equality is from the summability of
$H^{1}$ functions. The estimate for $||h\mathcal{L}_{h}(\pi_{sz}v)||_{\Omega}$
is then obtained by adding the potential term.

∎

Now we show that the discrete problem is well-posed. Let

(3.12)
$|||(v_{h},w_{h})|||^{2}=h^{-2\alpha}(v_{h},v_{h})_{\omega}+s_{h}(v_{h},v_{h})+s_{h}^{*}(w_{h},w_{h}).$

Define the discrete bilinear form

(3.13) $\displaystyle A[(u_{h},z_{h}),(v_{h},w_{h})]:=$ $\displaystyle
h^{-2\alpha}(u_{h},v_{h})_{\omega}+s_{h}(u_{h},v_{h})+a(v_{h},z_{h})$
$\displaystyle+a(u_{h},w_{h})-s_{h}^{*}(z_{h},w_{h}).$

######  Proposition 3.3.

BNBA Linear system (3.3) is well-posed. In fact, the following inf-sup
condition holds

$\underset{(v_{h},w_{h})\in V_{h}^{p}\times
W_{h}^{p}}{\sup}\frac{A_{h}[(u_{h},z_{h}),(v_{h},w_{h})]}{|||(v_{h},w_{h})|||}\geq|||(u_{h},z_{h})|||.$

###### Proof.

Let $v_{h}=u_{h}$, $w_{h}=-z_{h}$ we have

$\displaystyle A_{h}[(u_{h},z_{h}),(u_{h},-z_{h})]$
$\displaystyle=h^{-2\alpha}||u_{h}||^{2}_{\omega}+s_{h}(u_{h},u_{h})+s_{h}^{*}(z_{h},z_{h})$
$\displaystyle=|||(u_{h},z_{h})|||^{2}.$

∎

BNBA gives the stability of the linear system, now we give an upper bound of
the stabilizers.

######  Proposition 3.4.

boundA Let $[(u_{h},z_{h}),(v_{h},w_{h})]\in[V_{h}^{p}\times W_{h}^{p}]^{2}$,
$0\leq\alpha\leq 1$ and $\eta,\tau$ non-negative. We have the following bound

$s_{h}(u_{h},v_{h})+s^{*}_{h}(z_{h},w_{h})\leq
C(P)\left(||u_{h}||_{H^{1}(\Omega)}||v_{h}||_{H^{1}(\Omega)}+||z_{h}||_{H^{1}(\Omega)}||w_{h}||_{H^{1}(\Omega)}\right),$

where $C(P)=C(1+h^{2}||P||^{2}_{L^{\infty}}).$

###### Proof.

Consider each component of $s_{h}$. First we have by (3.9) and (3.11)

(3.14) $\displaystyle\mathcal{J}_{h}(u_{h},v_{h})$ $\displaystyle\leq
C\sum_{K}(h||u_{h}||_{H^{2}(K)}+||u_{h}||_{H^{1}(K)})(h||v_{h}||_{H^{2}(K)}+||v_{h}||_{H^{1}(K)})$
$\displaystyle\leq C||u_{h}||_{H^{1}(\Omega)}||v_{h}||_{H^{1}(\Omega)}$

Similarly for the second component we also have

(3.15) $h^{2}(\mathcal{L}_{h}u_{h},\mathcal{L}_{h}v_{h})_{\Omega}\leq
C(1+h^{2}||P||^{2}_{L^{\infty}})||u_{h}||_{H^{1}(\Omega)}||v_{h}||_{H^{1}(\Omega)}.$

Since $W_{h}^{p}\subset V_{h}^{p}$, we have by (3.14) and (3.15)

(3.16) $h^{2\eta}\mathcal{J}_{h}(z_{h},w_{h})\leq
Ch^{2\eta}||z_{h}||_{H^{1}(\Omega)}||w_{h}||_{H^{1}(\Omega)},$

and

(3.17) $h^{2\eta}(h\mathcal{L}_{h}z_{h},h\mathcal{L}_{h}w_{h})_{\Omega}\leq
C(1+h^{2}||P||^{2}_{L^{\infty}})h^{2\eta}||z_{h}||_{H^{1}(\Omega)}||w_{h}||_{H^{1}(\Omega)}.$

Again, by (3.10), we have

(3.18)
$h^{2\eta}\int_{\partial\Omega}h\partial_{n}z_{h}\partial_{n}w_{h}dS\leq
Ch^{2\eta}||z_{h}||_{H^{1}(\Omega)}||w_{h}||_{H^{1}(\Omega)},$

and by Cauchy–Schwarz inequality,

(3.19) $h^{\tau}(\nabla z_{h},\nabla w_{h})_{\Omega}\leq
h^{\tau}||z_{h}||_{H^{1}(\Omega)}||w_{h}||_{H^{1}(\Omega)}.$

Combining (3.14), (3.15), (3.16), (3.17), (3.18) and (3.19), we have the
desired estimate.

∎

##  4\. Stability and error analysis

In this section we discuss the error and stability estimate for the stabilized
scheme (3.3). This will be separated into two cases: $u\in H^{s}(\Omega)$ for
$1\leq s<2$ and $s\geq 2$.

###  4.1. Convergence analysis for solutions with low regularity

In this part we assume $u\in H^{s}(\Omega)$ with $1\leq s<2$ and $f\in
H^{-1}(\Omega)$. Then we prove the solution of (3.3) converges in this case.
Our aim is to show the discrete solution has an optimal estimate
$||u_{h}-u||_{H^{1}(\Omega)}\lesssim h^{s-1}||u||_{H^{s}(\Omega)}$. First, we
show the stabilizing term $s_{h}(\pi_{sz}u,v_{h})-G(f,v_{h})$ is bounded by
$||u||_{H^{s}(\Omega)}$ and $|||v_{h}|||$.

######  Lemma 4.1.

consis1 Let $u\in H^{s}(\Omega)$ with $1\leq s<2$. $\pi_{sz}u$ be the Scott-
Zhang interpolant of $u$ on $V^{p}_{h}$. Then $\forall v_{h}\in V^{p}_{h}$, we
have for some constant $C$ such that

$s_{h}(\pi_{sz}u,v_{h})-G(f,v_{h})\leq
C(1+h||P||_{L^{\infty}})h^{s-1}||u||_{H^{s}(\Omega)}\cdot|||v_{h}|||$

###### Proof.

First we have

$\displaystyle s_{h}(\pi_{sz}u,v_{h})$
$\displaystyle=\mathcal{J}_{h}(\pi_{sz}u,v_{h})+h^{2(s-1)}\langle\pi_{sz}u,v_{h}\rangle_{\Omega}+h^{2}(\mathcal{L}_{h}\pi_{sz}u,\mathcal{L}_{h}v_{h})_{\Omega}.$

by weakcons, we have

$\mathcal{J}_{h}(\pi_{sz}u,v_{h})\leq
Ch^{s-1}||u||_{H^{s}(\Omega)}(\mathcal{J}_{h}(v_{h},v_{h}))^{\frac{1}{2}}.$

and

$\displaystyle h^{2}(\mathcal{L}_{h}\pi_{sz}u,\mathcal{L}_{h}v_{h})_{\Omega}$
$\displaystyle\leq||h\mathcal{L}_{h}\pi_{sz}u||_{\Omega}||h\mathcal{L}_{h}v_{h}||_{\Omega}$
$\displaystyle\leq
C(1+h||P||_{L^{\infty}})h^{s-1}||u||_{H^{s}(\Omega)}||h\mathcal{L}_{h}v_{h}||_{\Omega}.$

Now since

$G(f,v_{h})=h^{2}(f_{h},\mathcal{L}_{h}v_{h})_{\Omega}\leq||hf_{h}||_{\Omega}||h\mathcal{L}_{h}v_{h}||_{\Omega},$

we only need to estimate $||hf_{h}||_{\Omega}$. Recall that $f_{h}\in
W_{h}^{p}$, we have $\forall w_{h}\in W^{p}_{h}$

$\displaystyle h(f_{h},w_{h})=$ $\displaystyle ha(u,w_{h})$ $\displaystyle=$
$\displaystyle h\int_{\Omega}\nabla(u-\pi_{sz}u)w_{h}+P(u-\pi_{sz}u)w_{h}dx$
$\displaystyle+h\int_{\Omega}\nabla\pi_{sz}u\cdot\nabla
w_{h}+P\pi_{sz}uw_{h}dx$ $\displaystyle\leq$ $\displaystyle
C(1+h^{2}||P||_{L^{\infty}})h^{s-1}||u||_{H^{s}(\Omega)}||w_{h}||_{\Omega}$
$\displaystyle+h\sum_{F}\int_{F}\llbracket\nabla\pi_{sz}u\cdot n\rrbracket
w_{h}dS+h(\mathcal{L}_{h}\pi_{sz}u,w_{h})_{\Omega}$ $\displaystyle\leq$
$\displaystyle
C(1+h||P||_{L^{\infty}})h^{s-1}||u||_{H^{s}(\Omega)}||w_{h}||_{\Omega},$

where the first inequality is due to optimal approximation property of Scott-
Zhang interpolant and the second inequality is obtained by weakcons and
applying discrete trace inequality (3.10). Finally by the definition of
$|||v_{h}|||$ we obtain the desired result.

∎

Now we give the second lemma which gives an estimate for the difference
between $\pi_{sz}u$ and the discrete solution $u_{h}$ under $|||\cdot|||$
norm.

######  Lemma 4.2.

stabnormest1 Let $(u_{h},z_{h})$ be the solution to (3.3), $u\in
H^{s}(\Omega)$ with $1\leq s<2$ be the exact weak solution to the continuous
problem, $0\leq\alpha\leq 1$ and $\eta\tau=0$. Then there exists some constant
$C$, such that

$|||(u_{h}-\pi_{sz}u,z_{h})|||\leq
C(1+h||P||_{L^{\infty}})h^{s-1}||u||_{H^{s}(\Omega)}.$

###### Proof.

We have from the $\inf-\sup$ condition that we only need to bound

$A[(u_{h}-\pi_{sz}u,z_{h}),(v_{h},z_{h})]\leq
Ch^{s-1}||u||_{H^{s}(\Omega)}|||(v_{h},w_{h})|||.$

Now we have that

(4.1) $\displaystyle A[(u_{h}-\pi_{sz}u,z_{h}),(v_{h},w_{h})]=$ $\displaystyle
h^{-2\alpha}(u-\pi_{sz}u,v_{h})_{\omega}$
$\displaystyle+a(u-\pi_{sz}u,w_{h})-s(\pi_{sz}u,v_{h})+G(f,v_{h}).$

For the first term, we have

(4.2) $h^{-2\alpha}(u-\pi_{sz}u,v_{h})_{\omega}\leq
Ch^{s+1-\alpha}||u||_{H^{s}(\Omega)}h^{-\alpha}||v_{h}||_{\omega}.$

The second and third term are bounded according to consis1, and for the last
term in (4.1), if $\eta=0$ we have

(4.3) $\displaystyle a(u-\pi_{sz}u,w_{h})=$
$\displaystyle\int_{\Omega}\nabla(u-\pi_{sz}u)\cdot\nabla
w_{h}dx+\int_{\Omega}P(u-\pi_{sz}u)w_{h}dx$ $\displaystyle\leq$
$\displaystyle\sum_{F}\int_{F}(u-\pi_{sz}u)\llbracket\nabla w_{h}\cdot
n\rrbracket dS$
$\displaystyle+\int_{\partial\Omega}(u-\pi_{sz}u)\partial_{n}w_{h}dS+(u-\pi_{sz}u,\mathcal{L}_{h}w_{h})_{\Omega}$
$\displaystyle\leq$ $\displaystyle
C||u-\pi_{sz}u||_{H^{1}(\Omega)}(s^{*}(w_{h},w_{h}))^{\frac{1}{2}}$
$\displaystyle\leq
Ch^{s-1}||u||_{H^{s}(\Omega)}(s^{*}(w_{h},w_{h}))^{\frac{1}{2}},$

where the second estimate is by (3.9) and the third estimate is by the optimal
convergence property of the Scott-Zhang interpolant. If instead, $\tau=0$,
then by the optimal convergence property of Scott-Zhang interpolant again we
have

$\displaystyle a(u-\pi_{sz}u,w_{h})$ $\displaystyle\leq
C(1+h||P||_{L^{\infty}})h^{s-1}||u||_{H^{s}(\Omega)}||w_{h}||_{H^{1}(\Omega)}$
$\displaystyle\leq
C(1+h||P||_{L^{\infty}})h^{s-1}||u||_{H^{s}(\Omega)}(s^{*}(w_{h},w_{h}))^{\frac{1}{2}}$

Then by combining the estimates for each term , we conclude that the estimate
in the lemma holds.

∎

From stabnormest1 we can get the boundedness of the discrete solution $u_{h}$.

######  Corollary 4.3.

bdness1 Let $u\in H^{s}(\Omega)$ with $1\leq s<2$, then

$||u_{h}||_{H^{1}(\Omega)}\leq C(1+h||P||_{L^{\infty}})||u||_{H^{s}(\Omega)},$

and

$h^{\frac{\tau}{2}}||z_{h}||_{\Omega}\leq
C(1+h||P||_{L^{\infty}})h^{s-1}||u||_{H^{s}(\Omega)}.$

###### Proof.

Since

$\displaystyle||u-u_{h}||_{H^{1}(\Omega)}$
$\displaystyle\leq||u-\pi_{sz}u||_{H^{1}(\Omega)}+||\pi_{sz}u-u_{h}||_{H^{1}(\Omega)}$
$\displaystyle\leq||u||_{H^{1}(\Omega)}+h^{-(s-1)}|||(\pi_{sz}u-u_{h},0)|||$
$\displaystyle\leq C(1+h||P||_{L^{\infty}})||u||_{H^{s}(\Omega)},$

we have $||u_{h}||_{H^{1}(\Omega)}\leq
C(1+h||P||_{L^{\infty}})||u||_{H^{s}(\Omega)}$. Similarly, we have

$h^{\frac{\tau}{2}}||z_{h}||_{\Omega}\leq|||(u-\pi_{sz}u,z_{h})|||\leq
C(1+h||P||_{L^{\infty}})h^{s-1}||u||_{H^{s}(\Omega)}.$

∎

Now we derive the weak convergence theorem for system (3.3) with $u\in
H^{1}(\Omega)$ and optimal convergence for $u\in H^{s}(\Omega)$ when $1\leq
s<2$ separately.

######  Theorem 4.4.

cvg Let $u\in H^{1}(\Omega)$, $s=1$, $0<\alpha\leq 1$, $\eta=0$, and $\tau>0$.
Then the stabilized finite element approximation $u_{h}$ has a subsequence
converging weakly to $u$ and we have the following estimate:
(1) $||u_{h}-q||_{\omega}\leq Ch^{\alpha}||u||_{H^{1}(\Omega)},$
(2) $||\mathcal{L}u_{h}-f||_{H^{-2}(\Omega)}\leq
C(h+h^{\frac{\tau}{2}})||u||_{H^{1}(\Omega)}$, where $C$ is some constant
depending on the potential $P$.

###### Proof.

It is shown in bdness1 that $||u_{h}||_{H^{1}(\Omega)}$ is bounded, so $u_{h}$
has a subsequence, still denoted by $u_{h}$, such that $u_{h}\rightharpoonup
v$ for some $v\in H^{1}(\Omega)$. We want to show for any $w\in
C_{0}^{\infty}(\Omega)$, $v$ satisfies

$(v,w)_{\omega}=(u,w)_{\omega}\;\;\text{ and }\;\;a(v,w)=a(u,w),$

and the uniqueness of the continuous problem to show the weak convergence of
$u_{h}$ to $u$. Recall that the discrete system reads

$\left\\{\begin{array}[]{rl}h^{-2\alpha}(u_{h},v_{h})_{\omega}+s_{h}(u_{h},v_{h})+a(v_{h},z_{h})&=h^{2\alpha}(q,v_{h})_{\omega}\\\
a(u_{h},w_{h})-s_{h}^{*}(z_{h},w_{h})&=<f,w_{h}>_{\Omega}\end{array}\right.,\;\forall(v_{h},w_{h})\in
V_{h}\times W_{h}.$

First, we estimate the residual $a(u_{h}-u,w)$. Let $w_{h}$ be the Scott-Zhang
interpolant of $w$ we have $a(u_{h}-u,w)=a(u_{h}-u,w-w_{h})+a(u_{h}-u,w_{h})$.
For the second term, we have

$\displaystyle a(u_{h}-u,w_{h})$ $\displaystyle=s^{*}(z_{h},w_{h})$
$\displaystyle\leq(s_{h}^{*}(z_{h},z_{h}))^{\frac{1}{2}}(s_{h}^{*}(w_{h},w_{h}))^{\frac{1}{2}}$
$\displaystyle\leq
C||u||_{H^{1}(\Omega)}(\mathcal{J}_{h}(w_{h},w_{h})+\int_{\partial\Omega}h\partial_{n}w_{h}\partial_{n}w_{h}dS$
$\displaystyle+h^{2}(\mathcal{L}_{h}w_{h},\mathcal{L}_{h}w_{h})_{\Omega}+h^{\tau}||w_{h}||_{H^{1}(\Omega)})^{\frac{1}{2}}$
$\displaystyle\leq
C(h+h^{\frac{\tau}{2}})||u||_{H^{1}(\Omega)}||w||_{H^{2}(\Omega)}.$

Note that we use the fact that the Scott-Zhang operator is bounded with
respect to the $H^{2}$-norm in the last inequality. For the first term, we
have

$\displaystyle a(u_{h}-u,w-w_{h})\leq
C||u_{h}-u||_{H^{1}(\Omega)}||w-w_{h}||_{H^{1}(\Omega)}\leq
Ch||u||_{H^{1}(\Omega)}||w||_{H^{2}(\Omega)}.$

Now for $(u_{h}-u,w)_{\omega}$, we have

$h^{-2\alpha}(u_{h}-\pi_{sz}u,w)_{\omega}\leq|||(u-\pi_{sz}u,0)|||\cdot||h^{-\alpha}w||_{\omega}\leq
Ch^{-\alpha}||u||_{H^{1}(\Omega)}||w||_{\omega},$

and

$(u-\pi_{sz}u,w)_{\omega}\leq Ch||u||_{H^{1}(\Omega)}|||w||_{\omega},$

which gives

$(u_{h}-u,w)_{\omega}\leq
C(h+h^{\alpha})||u||_{H^{1}(\Omega)}|||w||_{\omega}.$

∎

Notice that we derived a weak convergence result from cvg without specifying
$\alpha$, and $\tau$. However, different parameters may infect the condition
number of the discrete system and the convergence rate with respect to
$||u_{h}-q||_{\omega}$ and $||\mathcal{L}u_{h}-f||_{H^{-2}(\Omega)}$ as
indicated in cvg.

######  Theorem 4.5.

H-2norm Given that all parameters satisfying conditions in cvg, denote the
convergence rate of $||\mathcal{L}u_{h}-f||_{H^{-2}(\Omega)}$ with
$\lambda=\frac{\tau}{2}\leq 1$, i.e.

$||\mathcal{L}u_{h}-f||_{H^{-2}(\Omega)}\leq
Ch^{\lambda}||u||_{H^{1}(\Omega)}.$

Then, the discrete equations (3.3) has a condition number

$\mathcal{K}=\frac{C(P)}{h^{2\lambda+2}}.$

###### Proof.

First, we notice that $\alpha$ does not affect the condition number with
conditions in cvg because the largest singular value is bounded by
$\mathcal{J}_{h}$, which is $\frac{1}{h^{2}}$. Thus we set $\alpha=1$ to gain
an optimal convergence rate for $||u_{h}-u||_{\omega}$. In this case, the
largest singular value is $\frac{C(P)}{h^{2}}$ by the same argument as in
boundA. The smallest singular value is obtained by letting
$(v_{h},w_{h})=(u_{h},-z_{h})$. We have

$A[(u_{h},z_{h}),(u_{h},-z_{h})]=h^{-\alpha}(u_{h},u_{h})_{\omega}+s_{h}(u_{h},u_{h})+s^{*}_{h}(z_{h},z_{h})\geq
Ch^{\tau}||(u_{h},z_{h})||_{\Omega}^{2},$

by Poincaré inequality. In this case $\mathcal{K}=C(P)h^{-(2\lambda+2)}$.
∎

Now we derive an optimal convergence for the scheme with prior knowledge that
$u\in H^{s}(\Omega)$ with $1<s<2$.

######  Lemma 4.6.

H-1error12 Let $u\in H^{s}(\Omega)$ with $1<s<2$, $0\leq\alpha\leq 1$. $\tau$
and $\eta$ are non-negative and $\eta\tau=0$. Then the $H^{-1}$ residual
converges in optimal sense. That is to say, for $w\in H^{1}_{0}(\Omega)$,
define $r$ by $<r,w>=a(u-u_{h},w)$, then we have the estimate

$||u_{h}-u||_{\omega}+||r||_{H^{-1}(\Omega)}\leq
C(1+h||P||_{L^{\infty}})h^{s-1}||u||_{H^{s}(\Omega)}.$

###### Proof.

First we bound $||u_{h}-u||_{\omega}$. We have from stabnormest1

(4.4) $\displaystyle||u_{h}-u||_{\omega}$
$\displaystyle\leq||u_{h}-\pi_{sz}u||_{\omega}+||u-\pi_{sz}u||_{\omega}$
$\displaystyle\lesssim
h^{-\alpha}|||(u_{h}-\pi_{sz}u,0)|||+h^{s}||u||_{H^{s}(\Omega)}$
$\displaystyle\leq C(1+h||P||_{L^{\infty}})h^{s-\alpha}||u||_{H^{s}(\Omega}.$

Now for the residual, let $w_{h}$ be the Scott-Zhang interpolant of $w$, then
we have

$a(u_{h}-u,w)=a(u_{h}-u,w_{h})+a(u_{h}-\pi_{sz}u,w-w_{h})+a(\pi_{sz}u-u,w-w_{h}).$

We bound the three terms separately. First we have

(4.5) $\displaystyle a(u_{h}-u,w_{h})=s_{h}^{*}(z_{h},w_{h})$
$\displaystyle\leq C|||(0,z_{h})|||\cdot|||(0,w_{h})|||$ $\displaystyle\leq
C(1+h||P||_{L^{\infty}})|||(0,z_{h})|||\cdot||w||_{H^{1}(\Omega)},$

due to the boundedness of $|||\cdot|||$ in boundA and the boundedness of
Scott-Zhang operator. Then, for $a(u_{h}-\pi_{sz}u,w-w_{h})$, we have

(4.6) $\displaystyle a(u_{h}-\pi_{sz}u,w-w_{h})=$
$\displaystyle\int_{\Omega}\nabla(u_{h}-\pi_{sz}u)\cdot\nabla(w-w_{h})dx$
$\displaystyle+(P(u_{h}-\pi_{sz}u),w-w_{h})_{\Omega}$ $\displaystyle=$
$\displaystyle\sum_{K}\int_{K}(-\Delta+P)(u_{sz}-\pi_{h}u)(w-w_{h})dx$
$\displaystyle+\sum_{F}\int_{F}\llbracket\nabla(u_{h}-\pi_{sz}u)\cdot
n\rrbracket(w-w_{h})dS$ $\displaystyle\leq$ $\displaystyle
C||h\mathcal{L}_{h}(u_{h}-\pi_{sz}u)||_{\Omega}||h^{-1}(w-w_{h})||_{\Omega}$
$\displaystyle+(\mathcal{J}_{h}(u_{h}-\pi_{sz}u,u_{h}-\pi_{sz}u))^{\frac{1}{2}}||w||_{H^{1}(\Omega)}$
$\displaystyle\leq$ $\displaystyle
C|||(u_{h}-\pi_{sz}u,0)|||\cdot||w||_{H^{1}(\Omega)},$

and for $a(u-\pi_{sz}u,w-w_{h})$, we have

(4.7) $a(u-\pi_{sz}u,w-w_{h})\leq
C(1+h^{2}||P||_{L^{\infty}})h^{s-1}||u||_{H^{s}(\Omega)}||w||_{H^{1}(\Omega)}.$

By combining (4.4), (4.6), (4.5), (4.7) we get the desired result.

∎

######  Theorem 4.7.

prime12 Let $u\in H^{s}(\Omega)$ and $(u_{h},z_{h})$ be the solutions to
system (3.3). Let $0\leq\alpha\leq 1,\eta\tau=0$ and both non-negative. Let
$B$ be a region satisfying estimate in co2 and L2error; $\kappa,\epsilon$ be
the parameters therein. Then

$||u-u_{h}||_{H^{1}(B)}\leq C(P)h^{\kappa(s-1)}||u||_{H^{s}(\Omega)}$

and

$||u-u_{h}||_{B}\leq\exp\left(C\kappa^{1-\epsilon}(||P||^{\frac{2}{3}}_{L^{\infty}(\Omega)}+1)\right)h^{\kappa(s-1)}||u||_{H^{s}(\Omega)},$

###### Proof.

From conditional stability result we have for $u-u_{h}$, we have

$||u-u_{h}||_{H^{1}(B)}\leq
C(P)(||u-u_{h}||_{\omega}+||r||_{H^{-1}})^{\kappa}(||u-u_{h}||_{\Omega}+||r||_{H^{-1}})^{1-\kappa}.$

Note that from H-1error12, $||u-u_{h}||_{\omega}$ and $||r||_{H^{-1}}$
converge optimally. Bounding

$\displaystyle||u-u_{h}||_{\Omega}$ $\displaystyle\leq
Ch^{-(s-1)}|||(u-u_{h},z_{h})|||$ $\displaystyle\leq
C(1+h||P||_{L^{\infty}})||u||_{H^{s}(\Omega)},$

we obtain that

$||u-u_{h}||_{H^{1}(B)}\leq C(P)h^{\kappa(s-1)}||u||_{H^{s}(\Omega)}.$

Repeat the procedure we can also obtain the second estimate for $L^{2}$-norm
on $B$.

∎

###  4.2. Optimal convergence for solutions with higher regularity

In this part we assume $u$ enjoys higher regularity $s$, namely, $u\in
H^{s}(\Omega)$ for some $s\geq 2$, and $f\in H^{s-2}(\Omega)$. In this case,
we may assume $P$ has a higher regularity $P\in W^{s-2,\infty}(\Omega)$. The
discrete system (3.3) then reads:

(4.8)
$\left\\{\begin{array}[]{rl}h^{-2\alpha}(u_{h},v_{h})_{\omega}+s_{h}(u_{h},v_{h})+a(v_{h},z_{h})&=h^{-2\alpha}(\tilde{q},v_{h})_{\omega}+(h\tilde{f}_{h},h\mathcal{L}_{h}v_{h})\\\
a(u_{h},w_{h})-s_{h}^{*}(z_{h},w_{h})&=(\tilde{f}_{h},w_{h})_{\Omega}\end{array}\right.,$

where $u_{h}\in V_{h}^{p}$ and $w_{h}\in W_{h}^{p}$ are finite element spaces
with $p+1\geq s$. In this session, we will consider the convergence result
with perturbed data (at the discrete level), namely

(4.9) $\tilde{f}_{h}=f_{h}+\delta f,\;\;\;\tilde{q}=q+\delta q,$

where $\delta f,\delta q\in L^{2}(\Omega)$.

First we give a stability estimate of the linear system in terms of its
condition number.

######  Proposition 4.8.

con Let $0\leq\alpha\leq 1$, $\eta\geq 0$, and $0\leq\tau\leq 2(s-1)$. Then
the linear system (4.8) has a Euclidean condition number

$\mathcal{K}_{2}=C(1+h^{2}||P||_{L^{\infty}})h^{-2s}.$

###### Proof.

Let $A$ be as defined in (3.13). First We have for $\phi_{h},\psi_{h}\in
V^{p}_{h}$ or $W_{h}^{p}$,

$a(\phi_{h},\psi_{h})\leq\frac{C(1+h^{2}||P||^{2}_{L^{\infty}})}{h^{2}}||\phi_{h}||_{\Omega}||\psi_{h}||_{\Omega},$

by (3.11). Then from boundA we know that

$\sup_{\begin{subarray}{c}(u_{h},z_{h})\in V_{h}^{p}\times W_{h}^{p}\\\
(v_{h},w_{h})\in V_{h}^{p}\times
W_{h}^{p}\end{subarray}}\frac{A_{h}[(u_{h},z_{h}),(v_{h},w_{h})]}{||(u_{h},z_{h})||_{\Omega}||(v_{h},w_{h})||_{\Omega}}\leq
C\frac{1+h^{2}||P||^{2}_{L^{\infty}}}{h^{2}}.$

Moreover, from BNBA, we know that for some specific $(v_{h},w_{h})$, we have

$A_{h}[(u_{h},z_{h}),(v_{h},w_{h})]=||u_{h}||^{2}_{\omega}+s_{h}(u_{h},u_{h})+s_{h}^{*}(z_{h},z_{h}).$

Therefore, for all $(u_{h},z_{h})\in V_{h}^{p}\times W_{h}^{p}$, we have

$\sup_{(v_{h},w_{h})\in V_{h}^{p}\times
W_{h}^{p}}\frac{A_{h}[(u_{h},z_{h}),(v_{h},w_{h})]}{||(v_{h},w_{h})||_{\Omega}}\geq
Ch^{2(s-1)}||(u_{h},z_{h})||_{\Omega}.$

Now by [ern2006evaluation, Theorem 3.1] we have the desired estimate, where
$C(P)=C(1+h^{2}||P||^{2}_{L^{\infty}})$

∎

Our aim now is to show $u_{h}$ converges to $u$ in $L^{2}$-norm in $\omega$
and $H^{-1}$-norm in $\Omega$. The steps follow as the last section.

######  Lemma 4.9.

consis2 Let $\pi_{sz}u$ be the Scott-Zhang interpolant of $u$ on $V^{p}_{h}$.
Then $\forall v_{h}\in V^{p}_{h}$, we have for some constant $C$,

$\displaystyle s_{h}(\pi_{sz}u,v_{h})-$ $\displaystyle
h^{2}(f_{h},\mathcal{L}_{h}v_{h})_{\Omega}$ $\displaystyle\leq$ $\displaystyle
C\left((1+h^{2}||P||_{L^{\infty}})h^{s-1}||u||_{H^{s}(\Omega)}+h^{s-1}||f||_{H^{s-2}(\Omega)}\right)|||(v_{h},0)|||.$

###### Proof.

First we have

$\displaystyle s(\pi_{sz}u,v_{h})-h^{2}(f_{h},\mathcal{L}_{h}v_{h})=$
$\displaystyle\mathcal{J}_{h}(\pi_{sz}u,v_{h})+h^{2}(\mathcal{L}_{h}\pi_{sz}u-f_{h},\mathcal{L}_{h}v_{h})_{\Omega}$
$\displaystyle+h^{2(s-1)}\langle\pi_{sz}u,v_{h}\rangle_{\Omega}.$

The first quantity is bounded by using weakcons directly and the third
quantity is easily bounded by using Cauchy-Schwarz inequality. For the second
quantity, we have

$\displaystyle
h^{2}(\mathcal{L}_{h}\pi_{sz}u-f_{h},\mathcal{L}_{h}v_{h})_{\Omega}$
$\displaystyle=h^{2}(\mathcal{L}_{h}(\pi_{sz}u-u),\mathcal{L}_{h}v_{h})_{\Omega}+h^{2}(f-f_{h},\mathcal{L}_{h}v_{h})_{\Omega}$
$\displaystyle\leq
Ch^{s-1}\left(C(P)||u||_{H^{s}(\Omega)}+||f||_{H^{s-2}(\Omega)}\right)||h\mathcal{L}_{h}v_{h}||_{\Omega},$

where $C(P)=1+h^{2}||P||_{L^{\infty}}$. The estimate is from the fact that
$\pi_{sz}$ has an optimal convergence property for under all Sobolev norms and
$\pi_{h}$ has an optimal convergence property for $L^{2}$-norm
($f_{h}=\pi_{h}f$ when $f\in L^{2}(\Omega)$). Thus we obtain

$h^{2}(\mathcal{L}_{h}\pi_{sz}u-f_{h},\mathcal{L}_{h}v_{h})_{\Omega}\leq
C(1+h^{2}||P||_{L^{\infty}})h^{s-1}||u||_{H^{s}(\Omega)}||h\mathcal{L}_{h}v_{h}||_{\Omega}.$

Then by combining the two cases above we get the desired estimate

(4.10) $s_{h}(\pi_{sz}u,v_{h})-h^{2}(f_{h},\mathcal{L}_{h}v_{h})_{\Omega}\leq
C(1+h^{2}||P||_{L^{\infty}})h^{s-1}||u||_{H^{s}(\Omega)}|||(v_{h},0)|||.$

∎

######  Remark 4.10.

Notice that, comparing with consis1, consis2 requires the source term $f$ to
have a higher order regularity $f\in H^{s-2}(\Omega)$. This is not always the
case when the potential is only $L^{\infty}(\Omega)$. The cause of the problem
is that the second order consistency term
$h^{2}(f-f_{h},\mathcal{L}_{h}v_{h})$ may fail to generate higher order
convergence when $P$ (so as $f$) is rough.

######  Lemma 4.11.

stabnormest2 Let $(u_{h},z_{h})$ be the solution to (4.8), $u\in
H^{s}(\Omega)$ be the solution to the continuous problem. Let $\alpha\leq 1$,
and $\tau\eta=0$. Then

$|||(u_{h}-\pi_{sz}u,z_{h})|||\leq
C\left((1+h||P||_{L^{\infty}})h^{s-1}||u||_{H^{s}(\Omega)}+h^{s-1}||f||_{H^{s-2}(\Omega)}+Pt\right),$

where $Pt=h^{-\alpha}||\delta q||_{\omega}+h^{-\frac{\tau}{2}}||\delta
f||_{\Omega}$.

###### Proof.

The proof follows as stabnormest1. ∎

######  Corollary 4.12.

interiorest Let $\alpha\geq 0$. The discrete solution $u_{h}$ converges
optimally in $\omega$ without perturbation. More precisely,

$||u-u_{h}||_{\omega}\leq
C\left((1+h||P||_{L^{\infty}})h^{s-1}||u||_{H^{s}(\Omega)}+||\delta
q||_{\omega}+h^{-(\frac{\tau}{2}-\alpha)}||\delta f||_{\Omega}\right).$

###### Proof.

From the definition of $|||\cdot|||$ as (3.12), we have

$\displaystyle||u-u_{h}||_{\omega}$
$\displaystyle\leq||u-\pi_{sz}u||_{\omega}+||\pi_{sz}u-u_{h}||_{\omega}$
$\displaystyle\leq
Ch^{s-1}||u||_{H^{s}(\Omega)}+h^{\alpha}|||\pi_{sz}u-u_{h}|||$
$\displaystyle\leq
C\left((1+h||P||_{L^{\infty}})h^{s-1}||u||_{H^{s}(\Omega)}+||\delta
q||_{\omega}+h^{-(\frac{\tau}{2}-\alpha)}||\delta f||_{\Omega}\right).$

∎

Now, with the conditional stability estimates given in Section 2, we have the
following estimates

######  Theorem 4.13.

prime Let $u\in H^{s}(\Omega)$ be the solution to the continuous problem and
$(u_{h},z_{h})$ be the solutions to system (4.8). Let $0\leq\alpha\leq
1,\eta\tau=0$ and both non-negative. Let $B$ be a region satisfying estimate
in co2 and L2error; $\kappa,\epsilon$ be the parameters therein. Then

$||u-u_{h}||_{H^{1}(B)}\leq
C(P)h^{\kappa(s-1)}\left(||u||_{H^{s}(\Omega)}+Pt\right)$

and

$||u-u_{h}||_{B}\leq\exp\left(C\kappa^{1-\epsilon}(||P||^{\frac{2}{3}}_{L^{\infty}(\Omega)}+1)\right)h^{\kappa(s-1)}\left(||u||_{H^{s}(\Omega)}+Pt\right),$

where $Pt=h^{-(s-1)-\alpha}||\delta
q||_{\omega}+h^{-(s-1)-\frac{\tau}{2}}||\delta f||_{\Omega}$.

###### Proof.

See the proof of prime12 ∎

Note that when $s>1$, the choice of $\eta$ is robust. We could choose a set of
parameters that depend only on the regularity $s$ working for both lower order
and higher order solutions, namely, by letting $0<\alpha\leq 1,\eta=0$ and
$0<\tau\leq 2$. However, from prime, it is clear that the optimal choice for
$\alpha$ and $\tau$ is $\alpha=\tau=0$, to minimize the error caused by
perturbation. Now, if we control $\tau=0$, then any large number can be taken
for $\eta$ to generate convergence according to the above theorem.
Specifically, if $\eta\to\infty$, we got a reduced system, namely,
$s^{*}_{\eta}(z_{h},w_{h})=0$ and $s^{*}_{h}(z_{h},w_{h})=h^{\tau}\langle
z_{h},w_{h}\rangle_{\Omega}$. However we also note that this choice of
parameters doesn’t work for the low regularity case ($u\in H^{1}(\Omega)$)
because it violates the conditions posed in cvg that $\alpha>0$ and $\tau>0$.
We give a table for different parameters chosen for different smoothness of
$u$.

Table 1. Parameters range for both lower and higher order regularity solutions | $\alpha$ | $\eta$ | $\tau$ | Optimal parameters
---|---|---|---|---
$s=1$ | $0<\alpha\leq 1$ | $\eta=0$ | $0<\tau\leq 2$ | N/A
$s>1$ | $0\leq\alpha\leq 1$ | $\eta\geq 0,\eta\tau=0$ | $0\leq\tau\leq 2(s-1)$ | $\alpha=\tau=0$

Note that we have not given an optimal choice of parameters for lower
regularity case, because from H-2norm, there is a trade-off between
convergence rate and the stability of the linear system.

Now, we give an estimate for the interpolation error for numerical integration
or rounding error in computing the stiffness matrix of the linear system
(4.8). Denote the theoretical solution to (4.8) still by $u_{h}$ and the
computational solution by $\tilde{u}_{h}$. Let ($A_{h}$)$A_{h\bar{e}}$ be the
matrix (non)perturbed by rounding errors and numerical integration, and assume

(4.11) $||A_{h}-A_{h\bar{e}}||\leq\bar{e}||A_{h}||,$

By numerical linear algebra, with shape functions chosen to have mass matrix
$\mathcal{M}_{h}$ with condition number $\mathcal{K}(\mathcal{M}_{h})$, which
is uniform with $h$ due to quasi-uniformality of $\mathcal{T}_{h}$. Let
$U_{h}$ and $\tilde{U}_{h}$ be the coefficients of $u_{h}$ and $\tilde{u}_{h}$
in $V^{p}_{h}$, we have

(4.12) $\displaystyle||u_{h}-\tilde{u}_{h}||_{\Omega}$
$\displaystyle\leq\sigma^{\frac{1}{2}}_{max}(\mathcal{M})||U_{h}-\tilde{U}_{h}||_{N}$
$\displaystyle\leq\sigma^{\frac{1}{2}}_{max}(\mathcal{M})\mathcal{K}(A_{h})||U_{h}||_{N}$
$\displaystyle\leq\sigma^{\frac{1}{2}}_{max}(\mathcal{M})\mathcal{K}(A_{h})\sigma_{min}^{-\frac{1}{2}}(\mathcal{M})||M^{\frac{1}{2}}U_{h}||_{N}$
$\displaystyle\leq
C(1+h^{2}||P||^{2}_{L^{\infty}})\mathcal{K}^{\frac{1}{2}}(\mathcal{M}_{h})\frac{\bar{e}}{h^{2s}}||u_{h}||_{\Omega}.$

######  Corollary 4.14 (Error estimate with interpolation error).

err Let $\alpha=\tau=0$ and $\eta\to\infty$. Let $\bar{e}$ be the error with
respect to solving linear equations (rounding errors, interpolation errors,
etc), and $\tilde{u}_{h}$ be the solution considering $\epsilon$ , then by
(4.12):

$||u-\tilde{u}_{h}||_{H^{1}(B)}\leq
C(P)h^{\kappa(s-1)}\left(||u||_{H^{s}(\Omega)}+h^{-(s-1)}||\delta
g||_{\omega}+h^{-(s-1)}||\delta f||_{\Omega}\right)+R(h),$

where

$R(h)=C(P)\bar{e}\left(\frac{1}{h^{2s+1}}||u||_{H^{s}(\Omega)}+\frac{1}{h^{3s}}||\delta
f||_{\Omega}+\frac{1}{h^{3s}}||\delta q||_{\omega}\right),$

and
$C(P)=\exp\left(C\kappa^{1-\epsilon}(||P||^{\frac{2}{3}}_{L^{\infty}(\Omega)}+1)\right)$

###### Proof.

The additional error for $||u-\tilde{u}_{h}||_{H^{1}(B)}$ is
$||u_{h}-\tilde{u}_{h}||_{H^{1}(B)}$. Following inverse inequality (3.11) and
condition number in 4.12 we have

$\displaystyle||u_{h}-\tilde{u}_{h}||_{H^{1}(B)}$
$\displaystyle\leq\frac{C}{h}||u_{h}-\tilde{u}_{h}||_{\Omega}$
$\displaystyle\leq\frac{C(1+h^{2}||P||^{2}_{L^{\infty}})\bar{e}}{h^{2s+1}}||u_{h}||_{\Omega}$
$\displaystyle\leq\frac{C(1+h^{2}||P||^{2}_{L^{\infty}})\bar{e}}{h^{2s+1}}(||u_{h}-u||_{\Omega}+||u||_{\Omega})$
$\displaystyle\leq
C^{\prime}(P)\bar{e}(\frac{1}{h^{2s+1}}||u||_{H^{s}(\Omega)}+\frac{1}{h^{3s}}||\delta
f||_{\Omega}+\frac{1}{h^{3s}}||\delta q||_{\omega}),$

where
$C^{\prime}(P)=\exp\left(C\kappa^{1-\epsilon}(||P||^{\frac{2}{3}}_{L^{\infty}(\Omega)}+1)\right)$,
and the corollary follows.

∎

##  5\. Numerical experiments

We present here numerical experiments for Schrödinger unique continuation
problem (1.1). First, we will consider $\Omega$ to be the unit disk, and the
function to be a piecewise linear function which is not smooth at $y=0$.

######  Example 5.1.

H-1 Let $f\in H^{-1}(\Omega)$ to be defined as

$<f,w>_{\Omega}=\int_{-1}^{1}wdx,$

and consider the piece-wise linear function

$u_{0}(x,y)=\left\\{\begin{aligned} -y&&\texttt{$y>0$}\\\
0&&\texttt{else},\end{aligned}\right.$

It is easy too observe that $u_{0}$ is the solution to

$\left\\{\begin{aligned} -\Delta u=f&&\texttt{in $\Omega$}\\\
u=u_{0}&&\texttt{in $\omega$}.\end{aligned}\right.$

In this example we consider $\Omega$ to be the unit ball and the geometries we
are interested in to be

$\omega=\\{(x,y)\in\Omega|x>0,x^{2}+y^{2}>0.25\\}$

and

$B=\\{(x,y)\in\Omega|x>0,x^{2}+y^{2}>0.0625\\}$

Figure 1. Domain in H-1 Figure 2. Error comparison of schemes with $\eta=0$
and $\tau=0$

(a) $\eta=0,\alpha=1$ and $\tau\to\infty$. (b) $\eta\to\infty,\alpha=1$ and
$\tau=0$.

According to prime12, $\eta\tau=0$ is needed to generate optimal convergence
with respect to regularity $s$, but by cvg, $\eta=0$ is needed to generate
weak convergence for $H^{1}$-only case. We show by comparison that this does
strengthen the convergence. In Figure 2b it’s observed that the simplified
scheme by setting $s_{h}^{*}(z_{h},w_{h})=\langle z_{h},w_{h}\rangle_{\Omega}$
is generating way slower convergence in both $L^{2}$ and $H^{1}$-norm than the
scheme with dual stabilization terms defined in (3.5) with $\eta=0$. This
indicates that, from Table 1, the robustness for $\eta$ does not hold for
rough $u$.

Now we show the influence of parameter $\tau$ on the convergence rate. In this
example, we control $\alpha=1,\eta=0$.

Figure 3. Different parameters achieving increasing convergence

(a) $\tau=0$. (b) $\tau=0.5$. (c) $\tau=2$. (d) $\tau=10$.

In Figure 3d, we present an increase in convergence rate with respect to
$L^{2}$ and $H^{1}$ norm, which achieves their optima at $\frac{\tau}{2}=1$,
which shows consistency with cvg. $prs$ and $dus$ represents
$\sqrt{\mathcal{J}_{h}(u_{h},u_{h})}$ and $||z_{h}||_{\Omega}$. It is shown in
part (A) that $||z_{h}||_{\Omega}$ is bounded while $\tau=0$ and may not be
bounded when $\tau>0$, which is consistent with bdness1.

In the second example, we present numerical experiments for real Schrödinger
unique continuation problem (1.1) with the high order $H^{1}$-conforming
method (3.3). We will focus on the classical Hadamard example for ill-posed
elliptic equations with logarithm potential.

######  Example 5.2.

Consider the unique continuation problem

(5.1) $\left\\{\begin{array}[]{rll}-\Delta
u+10\log(y+\frac{1}{2})u&=10\log(y+\frac{1}{2})\sin{x}\sinh{y}&\text{in
$\Omega:(0,\pi)\times(0,1)$,}\\\ u(x,y)&=\sin(x)\sinh(y)&\text{for
$(x,y)\in\omega$}\end{array}\right.$

whose solution is given by

$u(x,y)=\sin(x)\sinh(y).$

For such Hadamard-type solutions, we consider the interior datum
$q=u|_{\omega}$ and study the convergence in the target set $B$ for two
geometric settings of $\omega$ and $B$, namely

(5.2)
$\omega=\Omega\backslash[\frac{\pi}{4},\frac{3\pi}{4}]\times[0.05,1],\;\;B=\Omega\backslash[\frac{\pi}{4},\frac{3\pi}{4}]\times[0.75,1],$

and

(5.3)
$\omega=(\frac{\pi}{4},\frac{3\pi}{4})\times(0.05,0.5),\;\;B=(\frac{\pi}{8},\frac{7\pi}{8})\times(0.05,0.75).$

Figure 4. Different domains.

(a) Domain (5.2).

(b) Domain (5.3).

In domain (5.2) we examined the $H^{1}$-error in the region of interest $B$
using three different orders of finite element space. It can be observed from
Figure 5b that the scheme with or without additional dual stabilizer
$s^{*}_{\eta}(\cdot,\cdot)$ (as defined in (3.5)) almost coincides, which
indicates the robustness of $\eta$ for higher regularity solutions.

Figure 5. Convergence comparison by letting $\eta=0$ and $\eta\to\infty$

(a) $\eta\to\infty$. (b) $\eta=0$.

Now we give a convergence comparison for different domains. In this example we
set $\alpha=\tau=0$ and $\eta\to\infty$.

Figure 6. Convergence comparison for different domains.

(a) Convergence in domain (5.2). (b) Convergence in domain (5.3).

In Figure 6b, we observe that for domain (5.2), the convergence rate for each
choice of order $p$ almost achieves optima, which means in prime,
$\kappa\approx 1$, while for domain (5.3), $\kappa$ is around $0.35$. It is
also worth noting that, for both cases, the finite element approximation using
$p=3$ space has a turning point for convergence while no perturbation of data
is forced. This is probably due to the ill-posedness (high condition number)
of the linear system system (4.8) and the enlarged quadratic error in
numerical integration by such ill-posedness, which is consistent with err.

##  Appendix A Proof of tbi

The proof is similar with [burman2020stabilized, Corollary 1]. Due to the
density of $C^{2}(\Omega)$ in $H^{2}(\Omega)$, it suffices to show the
inequality holds for $u\in C^{2}(\Omega)$. Let $r_{0}<r_{1}<r_{2}<r_{3}<r_{4}$
and $B_{j}\subset\Omega$, $j=0,1,2,3,4$. Choose $\rho(x)=-d(x,x_{0})$ and
$K=\bar{B_{4}}\backslash B_{0}$. Notice that $\rho$ has no critical point in
$K$. Let $\chi\in C_{0}^{\infty}(K)$ satisfies $\chi=1$ in $B_{3}\backslash
B_{1}$, and $w=\chi u$. Then according to Theorem car we have

$\displaystyle\int_{B_{4}\backslash B_{0}}(\tau^{3}|w|^{2}+\tau|\nabla
w|^{2})e^{2\tau\phi}dx$ $\displaystyle\leq C\int_{B_{4}\backslash
B_{0}}|\Delta w|^{2}e^{2\tau\phi}dx$ $\displaystyle\leq
2C\int_{B_{4}\backslash B_{0}}|-\Delta w+Pw|^{2}e^{2\tau\phi}dx$
$\displaystyle+2C\int_{B_{4}\backslash B_{0}}|Pw|^{2}e^{2\tau\phi}dx.$

Set $\tau_{0}=(4C||P||^{2}_{L^{\infty}(\Omega)})^{\frac{1}{3}}+1$, we obtain

(A.1) $\int_{B_{4}\backslash B_{0}}(\frac{\tau^{3}}{2}|w|^{2}+\tau|\nabla
w|^{2})e^{2\tau\phi}dx\leq 2C\int_{B_{4}\backslash B_{0}}|-\Delta
w+Pw|^{2}e^{2\tau\phi}dx.$

Denote $\Phi(r)=e^{-\alpha r}$, and then we can bound the left-hand side in
(A.1) by

$\int_{B_{2}\backslash B_{1}}(\tau^{3}|w|^{2}+\tau|\nabla
w|^{2})e^{2\tau\phi}dx\geq\tau
e^{2\tau\Phi(r_{2})}||u||^{2}_{H^{1}(B_{2})}-\tau
e^{2\tau}||u||^{2}_{H^{1}(B_{1})}$

since $\phi\leq 1$ and $\tau_{0}>1$. Moreover, the right-hand side can be
bounded by

$\displaystyle\int_{B_{4}\backslash B_{0}}|-\Delta
u+Pu|^{2}e^{2\tau\phi}dx+\int_{(B_{4}\backslash B_{3})\cup
B_{1}}|[\Delta,\chi]u|^{2}e^{2\tau\phi}dx$ $\displaystyle\leq
e^{2\tau}||-\Delta
u+Pu||^{2}_{L^{2}(\Omega)}+e^{2\tau\Phi(r_{3})}||u||^{2}_{H^{1}(\Omega)}+e^{2\tau}||u||^{2}_{H^{1}(B_{1})}.$

Combining the two bounds we obtain

(A.2) $\tau e^{2\tau\Phi(r_{2})}||u||^{2}_{H^{1}(B_{2})}\leq C\tau
e^{2\tau}(||u||^{2}_{H^{1}(B_{1})}+||-\Delta
u+Pu||^{2}_{L^{2}(\Omega)})+Ce^{2\tau\Phi(r_{3})}||u||^{2}_{H^{1}(\Omega)}.$

Then by setting $\tau_{0}=C(1+||P||^{\frac{2}{3}}_{L^{\infty}(\Omega)})$ for
some large $C$ to absorb the constant before $||u||_{H^{1}(B_{2})}$, we have

$||u||_{H^{1}(B_{2})}\leq
Ce^{\tau(1-\Phi(r_{2}))}(||u||_{H^{1}(B_{1})}+||-\Delta
u+Pu||_{L^{2}(\Omega)})+Ce^{\tau(\Phi(r_{3})-\Phi(r_{2}))}||u||_{H^{1}(\Omega)}.$

The conclusion follows from log by letting

$c=1-\Phi(r_{3}),\;\;\kappa=\frac{\Phi(r_{2})-\Phi(r_{3})}{1-\Phi(r_{3})}\in(0,1).$

##  Appendix B Proof of co2

To prove co2, we need the following Lemma

######  Lemma B.1.

aux Let $\Omega$ be any domain having smooth or piecewise polygonal boundary.
Then the auxiliary Schrödinger equation with complex Robin boundary is well-
posed, i.e, the system

$\left\\{\begin{array}[]{rcc}-\Delta u+Pu&=f&\texttt{in $\Omega$,}\\\
\partial_{n}u+iu&=0&\texttt{on $\partial\Omega$,}\end{array}\right.$

has a unique solution for $f\in(H^{1}(\Omega))^{\prime}$, and

$||u||_{H^{1}(\Omega)}\leq C(P)||f||_{(H^{1}(\Omega))^{\prime}}.$

Moreover, if $P$ is positive in $\Omega$, the constant is independent of $P$.

###### proof of aux.

To show existence and uniqueness of the problem, it suffices to show that if
the following primal and dual weak formulation has a solution then it’s the
unique solution. That is, there is a unique $u\in H^{1}(\Omega)$, such that
for all $v\in H^{1}(\Omega)$,

$\int_{\Omega}\nabla u\cdot\nabla\bar{v}dx+\int_{\Omega}Pu\bar{v}dx\pm
i\int_{\partial\Omega}u\bar{v}dS=\langle f,\bar{v}\rangle.$

To show this, consider the difference $w$ between two solutions, which
satisfies

$\int_{\Omega}\nabla w\cdot\nabla\bar{v}dx+\int_{\Omega}Pw\bar{v}dx\pm
i\int_{\partial\Omega}w\bar{v}dS=0.$

Let $v=w$. The real and imaginary part should be $0$, we deduce that $w=0$ on
the boundary, and so is its outer normal derivative. Since the right-hand side
is in $L^{2}(\Omega)$, we have that $w\in H^{2}(\Omega)$ by elliptic
regularity. See, for example [evans10]. Moreover, Since the normal derivatives
vanishes and the boundary is smooth. We have that $w\in H^{2}_{0}(\Omega)$.
According to car, we have for large $\tau$

$||w||_{H^{1}(\Omega)}\leq C\int_{\Omega}(\tau|\nabla
w|^{2}+\tau^{3}|w|^{2})dx\leq C\int_{\Omega}|\Delta w|^{2}dx=0.$

This shows the uniqueness of both problems, and thus the existence, and the
inverse bound follows by bounded inverse theorem, see, for example in [brezis,
Corollary 2.7]. In the specific case when $P\geq 0$, we choose $v=u$ in the
weak formulation and take real and imaginary part to get estimates

$||\nabla u||_{\Omega}\leq C||f||_{(H^{1}(\Omega))^{\prime}},$

and b

$||u||_{\partial\Omega}\leq C||f||_{(H^{1}(\Omega))^{\prime}}$

Combining the two equations and by Poincare’s inequality we have

$||u||_{H^{1}(\Omega)}\leq C||f||_{(H^{1}(\Omega))^{\prime}}.$

∎

##  Appendix C Proof of LABEL:L2error

Before proving the following estimate, we first state two inequalities for
semiclassical Soblev norm, see, for example, [dos2009limiting,
zworski2022semiclassical]. Let

$J^{s}=(1-\hbar^{2}\Delta)^{\frac{s}{2}}$

be the Bessel potential, we will use two estimates with semiclassical Sobolev
norm. Let $\theta$, $\eta\in C_{0}^{\infty}(\mathbb{R}^{n})$, where $\eta=1$
near supp($\theta$), and $A,B$ be two semiclassical pseudodifferential
operators of orders $s$, $m$, respectively. Then for all $p,q,N\in\mathbb{R}$,
there exists $C$, such that

(C.1) $||[A,B]v||_{H^{p}_{scl}}\leq C\hbar||v||_{H^{p+s+m-1}_{scl}},$ (C.2)
$||(1-\eta)A\theta v||_{H^{p}_{scl}}\leq C\hbar^{N}||v||_{H^{q}_{scl}}.$

To prove L2error, we need the following lemma:

######  Lemma C.1.

sft Let $v\in C_{0}^{\infty}(B(x_{0},R)\backslash B(x_{0},r))$ for some $R,r$,
$\hbar=\frac{1}{\tau}$, $F:=-\hbar^{2}e^{\frac{\phi}{\hbar}}\Delta
e^{\frac{\phi}{\hbar}}$ be a semiclassical pseudodifferential operator, where
$\tau$ and $\phi$ is defined as in car, and $P$ be the bounded potential.
Then, there exists a constant $C$, such that

$\sqrt{\hbar}||v||_{L^{2}}\leq C||(F+\hbar^{2}P)v||_{H^{-1}_{scl}},$

where $||\cdot||_{H^{-1}_{scl}}$ denotes semiclassical Sobolev norm, defined
by

$||u||_{H^{-1}_{scl}}:=||(1-\hbar^{2}\Delta)^{\frac{s}{2}}u||_{L^{2}(\mathbb{R}^{n})}$

###### Proof of sft.

A similar proof can be referred to [burman2020stabilized] for the convection-
diffusion equation, but we will give it for the readers’ convenience. Since
$v\in C_{0}^{\infty}(B(x_{0},R)\backslash B(x_{0},r))$, we construct the same
way as in tbi, and by Carleman estimate and some simple algebra, we have

$C\int_{\mathbb{R}^{n}}|e^{\frac{\phi}{\hbar}}\Delta
e^{-\frac{\phi}{\hbar}}v|^{2}dx\geq\int_{\mathbb{R}^{n}}(\hbar^{-1}-1)|\nabla
v|^{2}+(\hbar^{-3}-\hbar^{-2})v^{2}dx.$

Adding potential term on both sides and rescaling by $\hbar^{4}$ with
$\hbar<<1$ we obtain

$C\int_{\mathbb{R}^{n}}|(F+\hbar^{2}P)v|^{2}dx\geq\int_{\mathbb{R}^{n}}\hbar^{3}|\nabla
v|^{2}+(\hbar-C\hbar^{4}P^{2})v^{2}dx.$

Note that $||v||_{H^{1}_{scl}}=\int_{\mathbb{R}^{n}}\hbar^{2}|\nabla
v|^{2}+v^{2}dx$. By letting

(C.3) $\hbar^{3}<<\frac{1}{C||P||^{2}_{\infty}}$

the right-hand term with potential can be absorbed and thus we have

(C.4) $\sqrt{\hbar}||v||_{H^{1}_{scl}}\leq C||(F+\hbar^{2}P)v||_{L^{2}}$

Now let $\theta,\eta\in C_{0}^{\infty}(\mathbb{R}^{n})$ satisfying that
$\theta=1$ around support of $v$ and $\eta=1$ near support of $\theta$. Then
by (C.2), for $\hbar<\hbar_{0}$ for $\hbar_{0}$ depending only on geometry, we
have,

$\displaystyle||v||_{L^{2}}$ $\displaystyle=||J^{-1}v||_{H^{1}_{scl}}$
$\displaystyle\leq||\eta J^{-1}v||_{H^{1}_{scl}}+||(1-\eta)J^{-1}\theta
v||_{H^{1}_{scl}}$ $\displaystyle\leq C||\eta J^{-1}v||_{H^{1}_{scl}}$

Note that $\eta J^{-1}v$ is also compactly supported. By (C.4) we have

$\displaystyle\sqrt{\hbar}||v||_{L^{2}}\leq C\sqrt{\hbar}||\eta
J^{-1}v||_{H^{1}_{scl}}\leq C||(F+\hbar^{2}P)\eta J^{-1}v||_{L^{2}}$

Now commute $(F+\hbar^{2}P)$ and $\eta J^{-1}$, we have the estimate

$\displaystyle||[(F+\hbar^{2}P),\eta J^{-1}]v||_{L^{2}}$
$\displaystyle=||[F,\eta J^{-1}]v||_{L^{2}}+||[\hbar^{2}P,\eta
J^{-1}]v||_{L^{2}}$ $\displaystyle\leq
C\hbar||v||_{L^{2}}+C\hbar^{2}||P||_{L^{\infty}}||v||_{H^{-1}_{scl}},$

where the first estimate is from (C.2) and the second estimate is simply from
the definition of $||\cdot||_{H^{-1}_{scl}}$. Note that the constant $C$ above
onle depends on geometry, we can make $\hbar_{0}$ small enough to absorb the
first term, and by (C.3) the second term can also be absorbed. Thus

(C.5) $\sqrt{\hbar}||v||_{L^{2}}\leq
C||J^{-1}(F+\hbar^{2}P)v||_{L^{2}}=C||(F+\hbar^{2}P)v||_{H^{-1}_{scl}}.$

∎

###### Proof of L2error.

Let $r_{0}<r_{1,}<r_{1}$, $\chi\in C^{\infty}_{0}(B_{4}\backslash B_{0})$ and
$\chi=1$ in $B_{3}\backslash B_{1^{\prime}}$, $\psi\in
C^{\infty}_{0}(B_{1}\cup\Omega\backslash B_{2})$ and $\psi=1$ in
$(B_{4}\backslash B_{3})\cup(B_{1^{\prime}}\backslash B_{0})$, then we have
$\psi=1$ while $[F,\chi]\neq 0$. Let $u\in C^{\infty}(\Omega)$, and we obtain
by (C.2)

$||[F+\hbar^{2}P,\chi]e^{\frac{\phi}{\hbar}}u||_{H^{-1}_{scl}}\leq
C||[F,\chi]\psi e^{\frac{\phi}{\hbar}}u||_{H^{-1}_{scl}}\leq C\hbar||\psi
e^{\frac{\phi}{\hbar}}u||_{L^{2}}.$

Notice that $\chi e^{\frac{\phi}{\hbar}}u\in C_{0}^{\infty}(B_{4})$, which
satisfies the condition in sft. Thus combining (C.2) and (C.5) we obtain

$\displaystyle\sqrt{\hbar}||\chi e^{\frac{\phi}{\hbar}}u||_{L^{2}}$
$\displaystyle\leq C||(F+\hbar^{2}P)\chi
e^{\frac{\phi}{\hbar}}u||_{H^{-1}_{scl}}$ $\displaystyle\leq C\hbar^{2}||\chi
e^{\frac{\phi}{\hbar}}(-\Delta+P)u||_{H^{-1}_{scl}}+C\hbar||\psi
e^{\frac{\phi}{\hbar}}u||_{L^{2}}.$

Now according to the norm bound $||\cdot||_{H^{-1}_{scl}}\leq
C\hbar^{-2}||\cdot||_{H^{-1}}$ and the support of $\chi$ and $\psi$, a local
version of estimate is obtained:

(C.6)
$\sqrt{\hbar}||e^{\frac{\phi}{\hbar}}u||_{L^{2}(B_{3})}\leq\sqrt{\hbar}||e^{\frac{\phi}{\hbar}}u||_{L^{2}(B_{1})}+C||e^{\frac{\phi}{\hbar}}(-\Delta+P)u||_{H^{-1}(\Omega)}+C\hbar||\psi
e^{\frac{\phi}{\hbar}}u||_{L^{2}(\Omega)}.$

Notice the last term on the right-hand side can be separated into three
components, i.e. $B_{1},B_{3}\backslash B_{2},\Omega\backslash B_{3}$. Notice
that the $B_{1}$ part can be absorbed by
$\sqrt{\hbar}||e^{\frac{\phi}{\hbar}}u||_{L^{2}(B_{1})}$ and the
$B_{3}\backslash B_{2}$ part can be absorbed by the left-hand side for small
$\hbar$. Thus we have by (C.6),

$\displaystyle||u||_{L^{2}(B_{2})}\leq$ $\displaystyle
Ce^{-\frac{\Phi(r_{2})}{\hbar}}\left(||u||_{L^{2}(B_{1})}+\hbar^{-\frac{1}{2}}||(-\Delta+P)u||_{H^{-1}(\Omega)}\right)$
$\displaystyle+C\hbar^{-\frac{3}{2}}e^{-\frac{\Phi(r_{2})-\Phi(r_{3})}{\hbar}}||u||_{L^{2}(\Omega)}.$

Then by absorbing the polynomial term by exponential term and following the
same procedure as in tbi, the conclusion follows.

∎

## References