Error estimates for stabilized finite element methods applied to ill-posed problems

TL;DR

This paper develops error estimates for stabilized finite element methods applied to ill-posed problems, extending analysis to cases lacking coercivity or inf-sup stability, with numerical validation.

Contribution

It introduces a novel analysis framework for stabilized finite element methods on ill-posed problems without requiring coercivity or inf-sup conditions.

Findings

A priori error estimates derived for ill-posed problems

A posteriori error estimates established without coercivity assumptions

Numerical example confirms theoretical results

Abstract

We propose an analysis for the stabilized finite element methods proposed in, E. Burman, Stabilized finite element methods for nonsymmetric, noncoercive, and ill-posed problems. Part I: Elliptic equations. SIAM J. Sci. Comput., 35(6) 2013, valid in the case of ill-posed problems for which only weak continuous dependence can be assumed. A priori and a posteriori error estimates are obtained without assuming coercivity or inf-sup stability of the continuous problem. A numerical example illustrates the theory.