Efficient operator-coarsening multigrid schemes for local discontinuous Galerkin methods

TL;DR

This paper introduces an efficient multigrid scheme for local discontinuous Galerkin methods that coarsens flux operators separately, achieving optimal convergence without building full mesh hierarchies, applicable to complex geometries.

Contribution

The paper proposes a novel operator-coarsening multigrid approach for LDG discretizations that improves convergence and simplifies implementation on complex meshes.

Findings

Achieves optimal multigrid convergence rates in 2D and 3D tests.

Does not require explicit mesh hierarchy construction.

Effective on complex geometries and adaptive grids.

Abstract

An efficient $hp$-multigrid scheme is presented for local discontinuous Galerkin (LDG) discretizations of elliptic problems, formulated around the idea of separately coarsening the underlying discrete gradient and divergence operators. We show that traditional multigrid coarsening of the primal formulation leads to poor and suboptimal multigrid performance, whereas coarsening of the flux formulation leads to optimal convergence and is equivalent to a purely geometric multigrid method. The resulting operator-coarsening schemes do not require the entire mesh hierarchy to be explicitly built, thereby obviating the need to compute quadrature rules, lifting operators, and other mesh-related quantities on coarse meshes. We show that good multigrid convergence rates are achieved in a variety of numerical tests on 2D and 3D uniform and adaptive Cartesian grids, as well as for curved domains using implicitly defined meshes and for multi-phase elliptic interface problems with complex geometry. Extension to non-LDG discretizations is briefly discussed.