Stabilised finite element methods for ill-posed problems with conditional stability

TL;DR

This paper presents an analysis of an adjoint stabilised finite element method tailored for ill-posed problems with conditional stability, providing error estimates and numerical validation for the elliptic Cauchy problem.

Contribution

It introduces a stabilized finite element approach for ill-posed problems with conditional stability and offers a comprehensive numerical analysis including error estimates.

Findings

Error estimates are established for the method.

Numerical examples confirm theoretical results.

Applicable to elliptic Cauchy problems.

Abstract

In this paper we discuss the adjoint stabilised finite element method introduced in, E. Burman, Stabilized finite element methods for nonsymmetric, noncoercive and ill-posed problems. Part I: elliptic equations, SIAM Journal on Scientific Computing, and how it may be used for the computation of solutions to problems for which the standard stability theory given by the Lax-Milgram Lemma or the Babuska-Brezzi Theorem fails. We pay particular attention to ill-posed problems that have some conditional stability property and prove (conditional) error estimates in an abstract framework. As a model problem we consider the elliptic Cauchy problem and provide a complete numerical analysis for this case. Some numerical examples are given to illustrate the theory.