Agglomeration-Based Geometric Multigrid Solvers for Compact Discontinuous Galerkin Discretizations on Unstructured Meshes

TL;DR

This paper introduces a geometric multigrid solver for the Compact Discontinuous Galerkin method that efficiently handles unstructured meshes through agglomeration, improving solver performance for Poisson's equation.

Contribution

The paper develops a multigrid solver using agglomeration for arbitrary meshes, extendable to various DG methods, enhancing computational efficiency.

Findings

Excellent solver performance for Poisson's equation

Effective handling of arbitrary element shapes and dimensions

Compatibility with different DG discretizations

Abstract

We present a geometric multigrid solver for the Compact Discontinuous Galerkin method through building a hierarchy of coarser meshes using a simple agglomeration method which handles arbitrary element shapes and dimensions. The method is easily extendable to other discontinuous Galerkin discretizations, including the Local DG method and the Interior Penalty method. We demonstrate excellent solver performance for Poisson's equation, provided a flux formulation is used for the operator coarsening and a suitable switch function chosen for the numerical fluxes.