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

TL;DR

This paper develops a convergence theory for goal-oriented adaptive finite element methods applied to second-order semilinear elliptic equations, ensuring reliable approximation of specific quantities of interest.

Contribution

It introduces a goal-oriented adaptive finite element framework for semilinear problems and proves convergence with respect to the targeted quantity of interest.

Findings

Convergence of the method is theoretically established.

Numerical experiments confirm the theoretical predictions.

The approach extends existing linear problem techniques to semilinear cases.

Abstract

In this article we develop a convergence theory for goal-oriented adaptive finite element algorithms designed for a class of second-order semilinear elliptic equations. We briefly discuss the target problem class, and introduce several related approximate dual problems that are crucial to both the analysis as well as to the development of a practical numerical method. 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 this class of semilinear problems, and then describe some goal-oriented variations of the standard approach to AFEM (GOAFEM). Following the recent approach of Mommer-Stevenson and Holst-Pollock for increasingly general linear problems, we first establish a quasi-error contraction result for the primal problem. We then develop some additional estimates that make it possible to establish contraction of the combined primal-dual quasi-error, and subsequently show convergence with respect to the quantity of interest. Finally, a sequence of numerical experiments are then carefully examined. It is observed that the behavior of the implementation follows the predictions of the theory.