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 E. Burman, Stabilized finite element methods for nonsymmetric, noncoercive and ill-posed problems. Part I: elliptic equations, SIAM Journal on Scientific Computing [10] 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.

Unique continuation for the Helmholtz equation using stabilized finite element methods

“…For a detailed presentation of such discrete stabilizing operators we refer the reader to [5] or [7]. We define on V h and W h , respectively, the norms…”

“…When deriving error estimates for the stabilized FEM that we herein introduce, we shall make use of the mild condition kh 1. To keep down the technical detail we restrict the analysis to the case of piecewise affine finite element spaces, but the extension of the proposed method to the high order case follows using the stabilization operators suggested in [5] (see also [7] for a discussion of the analysis in the ill-posed case).…”

Data assimilation finite element method for the linearized Navier–Stokes equations in the low Reynolds regime

“…The order of the estimate depends on the Hölder coefficient of the continuous stability estimate which depends on the size of the measure domain and the distance between the target domain and the boundary of the computational domain. For simplicity, we restrict the discussion to piecewise affine continuous approximation spaces, but the arguments can be extended to higher order finite element spaces, with the expected improvement of convergence order, following the ideas of [20]. It should however be noted that the system matrix becomes increasingly ill-posed as the polynomial order increases and the computation becomes more sensitive to noise in the measured data, so the practical interest in using high order approximation spaces remains to be proven.…”

Primal-Dual Mixed Finite Element Methods for the Elliptic Cauchy Problem

“…Herein we will advocate a different approach based on discretization of the ill-posed physical model in an optimization framework, followed by regularization of the discrete problem. This primal-dual approach was first introduced by Burman in the papers [11,13,12,14], drawing on previous work by Bourgeois and Dardé on quasi reversibility methods [4,5,7,8] and further developed for elliptic data assimilation problems [17], for parabolic data reconstruction problems in [20,18] and finally for unique continuation for Helmholtz equation [19]. For a related method using finite element spaces with C 1 -regularity see [22] and for methods designed for well-posed, but indefinite problems, we refer to [9] and for second order elliptic problems on non-divergence form see [38] and [39].…”