Stabilised finite element methods for non-symmetric, non-coercive and ill-posed problems. Part I: elliptic equations

TL;DR

This paper introduces a stabilised finite element method for solving non-symmetric, indefinite, and ill-posed elliptic problems by simultaneously solving forward and adjoint problems, with proven optimal error estimates and numerical demonstrations.

Contribution

A novel stabilised finite element approach for non-symmetric indefinite problems, including error analysis and applicability to convection-diffusion and Helmholtz equations.

Findings

Optimal error estimates in $H^1$ and $L^2$ norms.

Method effectively stabilises non-symmetric indefinite problems.

Numerical examples demonstrate practical effectiveness.

Abstract

In this paper we propose a new method to stabilise non-symmetric indefinite problems. The idea is to solve a forward and an adjoint problem simultaneously using a suitable stabilised finite element method. Both stabilisation of the element residual and jumps of certain derivatives of the discrete solution over element faces may be used. Under the assumption of well posedness of the partial differential equation and its associated adjoint problem we prove optimal error estimates in $H^1$ and $L^2$ norms in an abstract framework. Some examples of problems that are neither symmetric nor coercive, but that enter the abstract framework are given. First we treat indefinite convection-diffusion equations, with non-solenoidal transport velocity and either pure Dirichlet conditions or pure Neumann conditions and then a Cauchy problem for the Helmholtz operator. Some numerical illustrations are given.