Optimal complexity of goal-oriented adaptive FEM for nonsymmetric linear elliptic PDEs

Abstract

We analyze a goal-oriented adaptive algorithm that aims to efficiently compute the quantity of interest $G(u^{\star })$ with a linear goal functional $G$ and the solution $u^{\star }$ to a general second-order nonsymmetric linear elliptic partial differential equation. The current state of the analysis of iterative algebraic solvers for nonsymmetric systems lacks the contraction property in the norms that are prescribed by the functional analytic setting. This seemingly prevents their application in the optimality analysis of goal-oriented adaptivity. As a remedy, this paper proposes a goal-oriented adaptive iteratively symmetrized finite element method (GOAISFEM). It employs a nested loop with a contractive symmetrization procedure, e.g., the Zarantonello iteration, and a contractive algebraic solver, e.g., an optimal multigrid solver. The various iterative procedures require well-designed stopping criteria such that the adaptive algorithm can effectively steer the local mesh refinement and the computation of the inexact discrete approximations. The main results consist of full linear convergence of the proposed adaptive algorithm and the proof of optimal convergence rates with respect to both degrees of freedom and total computational cost (i.e., optimal complexity). Numerical experiments confirm the theoretical results and investigate the selection of the parameters.

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