Quasi-matrix-free hybrid multigrid on dynamically adaptive Cartesian grids

TL;DR

This paper introduces a matrix-free, multigrid solver framework for dynamically adaptive Cartesian grids using spacetrees, combining geometric and algebraic multigrid techniques for robust elliptic problem solutions.

Contribution

It develops a hierarchical operator representation for multigrid on adaptive grids, enabling efficient, matrix-free, and parallelizable solvers with minimal memory footprint.

Findings

Supports arbitrary adaptive grids with efficient multilevel operations.

Achieves robust solutions for complex elliptic problems.

Enables parallel computation on manycore clusters.

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- 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.