Optimal convergence rates for goal-oriented FEM with quadratic goal functional

Abstract

We consider a linear elliptic PDE and a quadratic goal functional. The goal-oriented adaptive FEM algorithm (GOAFEM) solves the primal as well as a dual problem, where the goal functional is always linearized around the discrete primal solution at hand. We show that the marking strategy proposed in [M. Feischl, D. Praetorius and K. G. van der Zee, An abstract analysis of optimal goal-oriented adaptivity, SIAM J. Numer. Anal. 54 (2016), no. 3, 1423–1448] for a linear goal functional is also optimal for quadratic goal functionals, i.e., GOAFEM leads to linear convergence with optimal convergence rates.