Quasi-matrix-free Hybrid Multigrid on Dynamically Adaptive Cartesian Grids

Abstract

We present a family of spacetree-based multigrid realizations using the tree’s multiscale nature to derive coarse grids. They align with matrix-free geometric multigrid solvers as they never assemble the system matrices, which is cumbersome for dynamically adaptive grids and full multigrid. The most sophisticated realizations use BoxMG to construct operator-dependent prolongation and restriction in combination with Galerkin/Petrov-Galerkin coarse-grid operators. This yields robust solvers for nontrivial elliptic problems. We embed the algebraic, problem-dependent, and grid-dependent multigrid operators as stencils into the grid and evaluate all matrix-vector products in situ throughout the grid traversals. Such an approach is not literally matrix-free as the grid carries the matrix. We propose to switch to a hierarchical representation of all operators. Only differences of algebraic operators to their geometric counterparts are held. These hierarchical differences can be stored and exchanged with small memory footprint. Our realizations support arbitrary dynamically adaptive grids while they vertically integrate the multilevel operations through spacetree linearization. This yields good memory access characteristics, while standard colouring of mesh entities with domain decomposition allows us to use parallel many-core clusters. All realization ingredients are detailed such that they can be used by other codes.