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

Abstract

In this paper we discuss the adjoint stabilised finite element method introduced in Burman (SIAM J Sci Comput 35(6):A2752–A2780, 2013) 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.