Convergence of goal-oriented adaptive finite element methods for semilinear problems

In this article we develop convergence theory for a class of goal-oriented adaptive finite element algorithms for second order semilinear elliptic equations. We first introduce several approximate dual problems, and briefly discuss the target problem class. We then review some standard facts concerning conforming finite element discretization and error-estimate-driven adaptive finite element methods (AFEM). We include a brief summary of a priori estimates for semilinear problems, and then describe goal-oriented variations of the standard approach to AFEM (GOAFEM). Following the recent approach of Mommer-Stevenson and Holst-Pollock for linear problems, we then establish a contraction result for the primal problem. We then develop some additional estimates that make it possible to establish contraction of the combined quasi-error, and subsequently show convergence in the sense of the quantity of interest. Our analysis is based on the recent contraction frameworks for the semilinear problem developed by Holst, Tsogtgerel and Zhu and Bank, Holst, Szypowski and Zhu and those for linear problems as in Cascon, Kreuzer, Nochetto and Siebert, and Nochetto, Siebert and Veeser. In addressing the goal-oriented problem we base our framework on that of Mommer and Stevenson for symmetric linear problems and Holst and Pollock for nonsymmetric problems. Unlike the linear case, one must track linearized and approximate dual sequences in order to establish contraction with respect to the quantity of interest.