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

Erik Burman, Mingfei Lu, Lauri Oksanen

TL;DR

This work addresses the ill-posed unique continuation problem for the Schrödinger equation $-$\Delta u + Pu = f$ in $\Omega$ with interior data $u=q$ in $\omega$ by designing a parameterized stabilized finite element method that leverages a priori regularity. The authors derive Hölder-type conditional stability bounds from Carleman estimates and construct a discrete Lagrangian with primal/dual stabilization to achieve convergence aligned with these stability properties. They prove well-posedness of the discrete system and provide rigorous error estimates in low- and high-regularity regimes, including residual bounds in dual norms and perturbation-robust results. Numerical experiments validate the theory, explore parameter choices, and demonstrate the method’s ability to handle low-regularity solutions and data perturbations while maintaining stable convergence in interior regions.

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 \textit{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.

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

TL;DR

This work addresses the ill-posed unique continuation problem for the Schrödinger equation \Delta u + Pu = f\Omegau=q\omega$ by designing a parameterized stabilized finite element method that leverages a priori regularity. The authors derive Hölder-type conditional stability bounds from Carleman estimates and construct a discrete Lagrangian with primal/dual stabilization to achieve convergence aligned with these stability properties. They prove well-posedness of the discrete system and provide rigorous error estimates in low- and high-regularity regimes, including residual bounds in dual norms and perturbation-robust results. Numerical experiments validate the theory, explore parameter choices, and demonstrate the method’s ability to handle low-regularity solutions and data perturbations while maintaining stable convergence in interior regions.

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 \textit{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 -regularity is studied and the convergence rate for solutions with regularity higher than 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.
Paper Structure (10 sections, 27 theorems, 183 equations, 6 figures, 1 table)

This paper contains 10 sections, 27 theorems, 183 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Continuous stability estimate
  3. Stabilized finite element method
  4. Stability and error analysis
  5. Convergence analysis for solutions with low regularity
  6. Optimal convergence for solutions with higher regularity
  7. Numerical experiments
  8. Proof of \ref{['tbi']}
  9. Proof of \ref{['co2']} and \ref{['ultraweak']}
  10. Proof of \ref{['L2error']}

Key Result

Lemma 2.1

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 $\tau_0\ge 1$ and $C>0$ depending on $\alpha$ and $\rho$, such that for all $\tau\ge \tau_0$ and $w\in C^2_

Figures (6)

  • Figure 1: Domain in \ref{['H-1']}
  • Figure 2: Error comparison of schemes with $\eta = 0$ and $\tau = 0$
  • Figure 3: Different parameters achieving increasing convergence
  • Figure 4: Different domains.
  • Figure 5: Convergence comparison by letting $\eta = 0$ and $\eta\to \infty$
  • ...and 1 more figures

Theorems & Definitions (61)

  • Lemma 2.1: Carleman estimate
  • proof
  • Corollary 2.2: Three-ball inequality
  • proof
  • Remark 2.3
  • Corollary 2.4: Unique continuation property
  • proof
  • Corollary 2.5
  • proof
  • Corollary 2.6
  • ...and 51 more