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On inf-sup stability and optimal convergence of the quasi-reversibility method for unique continuation subject to Poisson's equation

Erik Burman, Mingfei Lu

TL;DR

This work addresses ill-posed Poisson-based unique continuation and Cauchy problems by developing a mixed quasi-reversibility discretization whose stability comes from inf-sup stable finite element space pairs rather than forward-problem stabilization. By carefully selecting trial and test spaces (including full conforming, reduced conforming, lifted Crouzeix–Raviart, and nonconforming variants) and defining residual-norm-based triple norms, the authors establish discrete stability, consistency, and optimal convergence rates in the $H^1$-sense, with $L^2$-convergence achievable via a stabilized residual-minimization approach that leverages conditional stability. They prove that, under appropriate inf-sup and consistency conditions, the discretization errors satisfy rates of the form $|||(e_h,oldsymbol{ extlambda}_h)||| \\le C(h^{s-1}+\epsilon^{1/2})||u||_{H^s}$, and discuss perturbation resilience and the impact of data perturbations. Numerical experiments corroborate the theoretical findings and illustrate how different space choices influence stability and convergence, including observed logarithmic and Hölder-type behavior in various regions. Overall, the paper links quasi-reversibility, primal-dual stabilization, and residual-minimization within the framework of conditional stability to enable robust, optimal discretizations of ill-posed continuation problems.

Abstract

In this paper, we develop a framework for the discretization of a mixed formulation of quasi-reversibility solutions to ill-posed problems with respect to Poisson's equations. By carefully choosing test and trial spaces a formulation that is stable in a certain residual norm is obtained. Numerical stability and optimal convergence are established based on the conditional stability property of the problem. Tikhonov regularisation is necessary for high order polynomial approximation, , but its weak consistency may be tuned to allow for optimal convergence. For low order elements a simple numerical scheme with optimal convergence is obtained without stabilization. We also provide a guideline for feasible pairs of finite element spaces that satisfy suitable stability and consistency assumptions. Numerical experiments are provided to illustrate the theoretical results.

On inf-sup stability and optimal convergence of the quasi-reversibility method for unique continuation subject to Poisson's equation

TL;DR

This work addresses ill-posed Poisson-based unique continuation and Cauchy problems by developing a mixed quasi-reversibility discretization whose stability comes from inf-sup stable finite element space pairs rather than forward-problem stabilization. By carefully selecting trial and test spaces (including full conforming, reduced conforming, lifted Crouzeix–Raviart, and nonconforming variants) and defining residual-norm-based triple norms, the authors establish discrete stability, consistency, and optimal convergence rates in the -sense, with -convergence achievable via a stabilized residual-minimization approach that leverages conditional stability. They prove that, under appropriate inf-sup and consistency conditions, the discretization errors satisfy rates of the form , and discuss perturbation resilience and the impact of data perturbations. Numerical experiments corroborate the theoretical findings and illustrate how different space choices influence stability and convergence, including observed logarithmic and Hölder-type behavior in various regions. Overall, the paper links quasi-reversibility, primal-dual stabilization, and residual-minimization within the framework of conditional stability to enable robust, optimal discretizations of ill-posed continuation problems.

Abstract

In this paper, we develop a framework for the discretization of a mixed formulation of quasi-reversibility solutions to ill-posed problems with respect to Poisson's equations. By carefully choosing test and trial spaces a formulation that is stable in a certain residual norm is obtained. Numerical stability and optimal convergence are established based on the conditional stability property of the problem. Tikhonov regularisation is necessary for high order polynomial approximation, , but its weak consistency may be tuned to allow for optimal convergence. For low order elements a simple numerical scheme with optimal convergence is obtained without stabilization. We also provide a guideline for feasible pairs of finite element spaces that satisfy suitable stability and consistency assumptions. Numerical experiments are provided to illustrate the theoretical results.
Paper Structure (20 sections, 22 theorems, 156 equations, 3 figures)

This paper contains 20 sections, 22 theorems, 156 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Model problems and stability of the unique continuation
  3. Model problem: unique continuation from data in the bulk
  4. Model problem: Cauchy problem
  5. Continuous stability
  6. The finite element formulation of quasi-reversibility
  7. Finite element preliminaries
  8. Discrete formulation: unique continuation
  9. Discrete formulation: Cauchy problem
  10. Stability and error analysis
  11. Optimal convergence under $L^2$-norm
  12. Stability under perturbation
  13. Feasible finite element spaces and stability analysis
  14. Full conforming spaces and stability
  15. Reduced conforming spaces in $d\ge3$
  16. ...and 5 more sections

Key Result

Lemma 2.2

Assume $u\in H^1(\Omega)$ solves (ucp) or (cp) and satisfies that Define the data measurement $d(u)$ for the unique continuation problem as: and for the Cauchy problem as Let $G\subset\Omega$ such that $\omega\subset G$ and $dist(G,\partial\Omega)>0$ for the unique continuation problem, or $dist(G, \Gamma_1)>0$ for the Cauchy problem. Then, there exists a constant $C>0$ and $\tau\in (0,1)$ depe

Figures (3)

  • Figure 1: Convergence rates for different test spaces and oscillation rates
  • Figure 2: Error comparison for different test spaces. Observe different scales.
  • Figure 3: Convergence rates comparison in the interior region $G$

Theorems & Definitions (50)

  • Remark 2.1
  • Lemma 2.2: Local conditional stability estimate
  • Lemma 2.3
  • Remark 2.4
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Proposition 3.5
  • proof
  • ...and 40 more