Generalized preconditioned conjugate gradients for adaptive FEM with optimal complexity

Abstract

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.