Comparison of algebraic multigrid methods for an adaptive space–time finite‐element discretization of the heat equation in 3D and 4D

Abstract

SummaryThe aim of this work is to compare algebraic multigrid (AMG) preconditioned GMRES methods for solving the nonsymmetric and positive definite linear systems of algebraic equations that arise from a space–time finite‐element discretization of the heat equation in 3D and 4D space–time domains. The finite‐element discretization is based on a Galerkin–Petrov variational formulation employing piecewise linear finite elements simultaneously in space and time. We focus on a performance comparison of conventional and modern AMG methods for such finite‐element equations, as well as robustness with respect to the mesh discretization and the heat capacity constant. We discuss different coarsening and interpolation strategies in the AMG methods for coarse‐grid selection and coarse‐grid matrix construction. Further, we compare AMG performance for the space–time finite‐element discretization on both uniform and adaptive meshes consisting of tetrahedra and pentachora in 3D and 4D, respectively. The mesh adaptivity occurring in space and time is guided by a residual‐based a posteriori error estimation.