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

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 and quasi-optimality results for a method of this type for second order Dirichlet problems involving the elliptic operator Lu = ∇ · (A∇u)− b · ∇u− cu, with A Lipschitz, almost-everywhere symmetric positive definite (SPD), 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 goaloriented variation of standard AFEM (GOAFEM). Following the recent work of Mommer and Stevenson for symmetric problems, we establish contraction of GOAFEM. We also then show 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 et. al, by Nochetto, Siebert, and Veeser, and by Holst, Tsogtgerel, and Zhu. In the last part of the paper we perform a complexity analysis, and establish quasi-optimal cardinality of GOAFEM. We include an appendix discussion of the duality estimate as we use it here in an effort to make the paper more self-contained.