Paula Hilbert

Thomas Führer, Paula Hilbert, Ani Miraçi et al.

We analyze optimal complexity of adaptive finite element methods (AFEMs) for general second-order linear elliptic partial differential equations (PDEs) in the Lax-Milgram setting. To this end, we formulate an adaptive algorithm which steers the local mesh-refinement as well as the termination of a generalized minimal residual solver (GMRES) with optimal preconditioner to solve the arising non-symmetric finite element systems. Algorithmic interplay of mesh-refinement and iterative solver is shown to be optimal: A natural and fully computable quasi-error monitoring discretization error and algebraic solver error guarantees unconditional convergence for any choice of adaptivity parameters, i.e., the algorithm cannot fail to converge. This is ensured algorithmically via a novel adaptive feedback-control for the solver-termination parameter that monitors and ensures full R-linear convergence. Finally, the quasi-error even decays with optimal rates with respect to the overall computational complexity if the adaptivity parameters are chosen sufficiently small.

Paula Hilbert, Ani Miraçi, Dirk Praetorius

We consider adaptive finite element methods (AFEMs) with inexact algebraic solvers for second-order symmetric linear elliptic diffusion problems. Optimal complexity of AFEM, i.e., optimal convergence rates with respect to the overall computational cost, hinges on two requirements on the solver. First, each solver step is of linear cost with respect to the number of degrees of freedom. Second, each solver step guarantees uniform contraction of the solver error with respect to the PDE-related energy norm. Both properties must be ensured robustly with respect to the local mesh size h (i.e., h-robustness). While existing literature on geometric multigrid methods (MG) or symmetric additive Schwarz preconditioners for the preconditioned conjugate gradient method (PCG) that are appropriately adapted to adaptive mesh-refinement satisfy these requirements, this paper aims to consider more general solvers. Our main focus is on preconditioners stemming from contractive solvers which need not be symmetrized to be used with Krylov methods and which are not only h-robust but also p-robust, i.e., the contraction constant is independent of the polynomial degree p. In particular, we show that generalized PCG (GPCG) with an h- and p-robust contractive MG as a preconditioner satisfies the requirements for optimal-complexity AFEM and that it numerically outperforms AFEM using MG as a solver. While this is certainly known for (quasi-)uniform meshes, the main contribution of the present work is the rigorous analysis of the interplay of the solver with adaptive mesh-refinement. Numerical experiments underline the theoretical findings.