Cost-optimal goal-oriented adaptive FEM with nested iterative solvers

Abstract

Based on (Bringmann, Brunner, Praetorius, Streitberger: 2024), this talk presents a cost-optimal goal-oriented adaptive FEM (GOAFEM) algorithm for the efficient computation of a goal value \(G(u^\star)\) for the solution \(u^\star\) to a nonsymmetric linear elliptic partial differential equation (PDE). The recent work (Bringmann, Feischl, Miraçi, Praetorius, Streitberger:2024) showed that the key to cost-optimality is full R-linear convergence of an appropriate quasi-error quantity together with optimal convergence rates with respect to the number of degrees of freedom. Therein, the contraction of an iterative solver in the PDE-related norm is a crucial ingredient in the analysis. While a natural candidate for nonsymmetric PDEs is a preconditioned generalized minimal residual (GMRES) method, it only leads to the contraction of the residual in a discrete vector norm, and the connection to the PDE-related norm is not clear. Therefore, we follow the approach of (Bringmann, Feischl, Miraçi, Praetorius, Streitberger:2024) and consider a nested iterative solver, where the outer solver is a symmetrization method (the so-called Zarantonello iteration) and the inner solver is an optimal geometric multigrid method (Innerberger, Miraçi, Praetorius, Streitberger: 2024) for the symmetrized problem. Following this approach, we show that an embedding of nested iterative solvers into the standard GOAFEM loop SOLVE & ESTIMATE — MARK — REFINE guarantees full R-linear convergence of an appropriate quasi-error product so that convergence rates with respect to the number of degrees of freedom and with respect to the total runtime coincide. Finally, we prove optimal complexity of the proposed algorithm for sufficiently small adaptivity parameters. Numerical experiments investigate the performance of the algorithm and indicate that larger stopping parameters are feasible and even favorable in practice.