Cost-optimal goal-oriented adaptive FEM for linear elliptic PDEs

Abstract

For a given bounded Lipschitz domain and given right-hand side f, we consider the nonsymmetric second-order linear elliptic PDE. We suppose that the PDE satisfies the assumptions of the Lax–Milgram lemma so that the weak formulation of the PDE admits a unique solution. In contrast to standard adaptive FEM that aims to approximate the exact solution on the whole domain, goal-oriented adaptive FEM aims at computing a linear quantity of interest with the help of the so-called dual problem seeks. We formulate and analyze a goal-oriented adaptive finite element algorithm that steers the adaptive mesh refinement, and the inexact iterative solutions of the arising linear systems. While the analysis for symmetric PDEs (with convection b = 0) is considerably less challenging, the iterative solver for the nonsymmetric problem employs, as an outer loop, the so-called Zarantonello iteration to symmetrize the primal and dual problem and, as an inner loop, an optimal geometric multigrid algorithm. We prove that the proposed goal-oriented adaptive iteratively symmetrized finite element method (GAISFEM) leads to full linear convergence and, as our main contribution, to optimal convergence rates with respect to the overall computational cost, i.e., the total computational time.