Convergence of Goal-Oriented Adaptive Finite Element Methods for Nonsymmetric Problems

TL;DR

This paper develops convergence theory for goal-oriented adaptive finite element methods applied to nonsymmetric elliptic problems, establishing contraction results and demonstrating numerical performance on convection problems.

Contribution

It extends convergence analysis of goal-oriented AFEM to nonsymmetric problems, providing new theoretical foundations and contraction results.

Findings

Proven contraction of GOAFEM for nonsymmetric problems.

Established convergence in terms of the goal function.

Numerical results show effective performance on convection problems.

Abstract

In this article we develop convergence theory for a class of goal-oriented adaptive finite element algorithms for second order nonsymmetric linear elliptic equations. In particular, we establish contraction results for a method of this type for Dirichlet problems involving the elliptic operator L u = div (A grad u) - (b,grad u) - cu, with A Lipschitz, almost-everywhere symmetric positive definite, with b divergence-free, and with c >= 0. We first describe the problem class and review some standard facts concerning conforming finite element discretization and error-estimate-driven adaptive finite element methods (AFEM). We then describe a goal-oriented variation of standard AFEM (GOAFEM). Following the recent work of Mommer and Stevenson for symmetric problems, we establish contraction of GOAFEM and convergence in the sense of the goal function. Our analysis approach is signficantly different from that of Mommer and Stevenson, combining the recent contraction frameworks developed by Cascon, Kreuzer, Nochetto and Siebert; by Nochetto, Siebert and Veeser; and by Holst, Tsogtgerel and Zhu. We include numerical results demonstrating performance of our method with standard goal-oriented strategies on a convection problem.