Efficient low-order refined preconditioners for high-order matrix-free continuous and discontinuous Galerkin methods

TL;DR

This paper introduces an efficient, parallelizable preconditioner for high-order finite element discretizations that leverages low-order refined meshes and sum factorization, improving solution robustness and efficiency.

Contribution

It develops a novel low-order refined preconditioner for matrix-free high-order DG methods, incorporating anisotropic mesh treatment and extending to interior penalty schemes.

Findings

Preconditioner is robust in $h$ and $p$

Achieves uniform convergence across various examples

Effective for both continuous and discontinuous Galerkin methods

Abstract

In this paper, we design preconditioners for the matrix-free solution of high-order continuous and discontinuous Galerkin discretizations of elliptic problems based on FEM-SEM equivalence and additive Schwarz methods. The high-order operators are applied without forming the system matrix, making use of sum factorization for efficient evaluation. The system is preconditioned using a spectrally equivalent low-order ($p=1$) finite element operator discretization on a refined mesh. The low-order refined mesh is anisotropic and not shape regular in the polynomial degree of the high-order operator, requiring specialized solvers to treat the anisotropy. We make use of an element-structured, geometric multigrid V-cycle with ordered ILU(0) smoothing. The preconditioner is parallelized through an overlapping additive Schwarz method that is robust in $h$ and $p$. The method is extended to interior penalty and BR2 discontinuous Galerkin discretizations, for which it is also robust in the size of the penalty parameter. Numerical results are presented on a variety of examples, verifying the uniformity of the preconditioner.