An adaptive multigrid solver for DPG methods with applications in linear acoustics and electromagnetics

TL;DR

This paper introduces an adaptive multigrid preconditioner tailored for DPG discretizations, effectively solving complex acoustic and electromagnetic problems, especially in high-frequency regimes, by integrating mesh refinement with iterative solvers.

Contribution

It presents a novel multigrid preconditioning approach that uses trace spaces on mesh skeletons, compatible with adaptive hp-meshes, and integrates automatic mesh refinement with iterative solvers.

Findings

Effective in high-frequency electromagnetic simulations

Reduces computational complexity for large-scale problems

Theoretically justified for uniform meshes

Abstract

We propose an adaptive multigrid preconditioning technology for solving linear systems arising from Discontinuous Petrov-Galerkin (DPG) discretizations. Unlike standard multigrid techniques, this preconditioner involves only trace spaces defined on the mesh skeleton, and it is suitable for adaptive hp-meshes. The key point of the construction is the integration of the iterative solver with a fully automatic and reliable mesh refinement process provided by the DPG technology. The efficacy of the solution technique is showcased with numerous examples of linear acoustics and electromagnetic simulations, including simulations in the high-frequency regime, problems which otherwise would be intractable. Finally, we analyze the one-level preconditioner (smoother) for uniform meshes and we demonstrate that theoretical estimates of the condition number of the preconditioned linear system can be derived based on well established theory for self-adjoint positive definite operators.