What question did this study set out to answer?

The aim is to establish a stable and convergent numerical framework for quasi-reversibility methods applied to Poisson’s equations.

January 18, 2026Open Access

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

Key Points

The aim is to establish a stable and convergent numerical framework for quasi-reversibility methods applied to Poisson’s equations.
Developed a mixed formulation of quasi-reversibility methods.
Carefully selected test and trial spaces to ensure stability.
Analyzed numerical stability based on conditional stability properties.
Incorporated Tikhonov regularization for high-order approximations.
Provided guidelines for feasible finite element space pairs.
Demonstrated optimal convergence without stabilization for low-order elements.
Confirmed that high-order polynomial approximation requires Tikhonov regularization.
Illustrated theoretical results through numerical experiments.

Abstract

In this paper, we develop a framework for the discretization of a mixed formulation of quasi-reversibility methods for 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 regularization 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 assumptions. Numerical experiments are provided to illustrate the theoretical results.

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