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 results for a method of this type for Dirichlet problems involving the elliptic operator Lu = del.(A del u) - b . del u - cu, with A Lipschitz, 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. Following the recent work of Mommer and Stevenson for symmetric problems, we establish contraction and convergence of the goal-oriented method 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. (C) 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 479-509, 2016