LFA-tuned matrix-free multigrid method for the elastic Helmholtz equation

TL;DR

This paper introduces a matrix-free geometric multigrid method for the elastic Helmholtz equation, optimizing discretization and solver compatibility to enable scalable solutions for large 3D problems.

Contribution

It develops a novel, discretization-compatible multigrid solver that is matrix-free and optimized via local Fourier analysis for elastic Helmholtz equations.

Findings

Validated discretization and solver compatibility with Fourier analysis

Achieved optimal convergence rates through stencil weight tuning

Demonstrated scalability for large 3D problems

Abstract

We present an efficient matrix-free geometric multigrid method for the elastic Helmholtz equation, and a suitable discretization. Many discretization methods had been considered in the literature for the Helmholtz equations, as well as many solvers and preconditioners, some of which are adapted for the elastic version of the equation. However, there is very little work considering the reciprocity of discretization and a solver. In this work, we aim to bridge this gap. By choosing an appropriate stencil for re-discretization of the equation on the coarse grid, we develop a multigrid method that can be easily implemented as matrix-free, relying on stencils rather than sparse matrices. This is crucial for efficient implementation on modern hardware. Using two-grid local Fourier analysis, we validate the compatibility of our discretization with our solver, and tune a choice of weights for the stencil for which the convergence rate of the multigrid cycle is optimal. It results in a scalable multigrid preconditioner that can tackle large real-world 3D scenarios.