Quasi-matrix-free hybrid multigrid on dynamically adaptive Cartesian grids
This work addresses the challenge of implementing efficient multigrid solvers on dynamically adaptive grids for nontrivial elliptic problems, offering a memory-efficient and parallelizable solution.
The paper presents a family of spacetree-based multigrid solvers that avoid assembling system matrices, using BoxMG for operator-dependent prolongation/restriction and hierarchical stencil representations to reduce memory footprint. The approach supports dynamically adaptive grids and parallel manycore clusters with good memory access.
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- and grid-dependent multigrid operators as stencils into the grid and evaluate all matrix-vector products in-situ throughout the grid traversals. While such an approach is not literally matrix-free---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 manycore clusters. All realization ingredients are detailed such that they can be used by other codes.