A performance comparison of continuous and discontinuous Galerkin methods with fast multigrid solvers

TL;DR

This paper compares the performance of continuous and discontinuous Galerkin finite element methods with advanced multigrid solvers, demonstrating that matrix-free implementations significantly outperform matrix-based solvers in speed, especially for high-order methods.

Contribution

It introduces a comprehensive performance comparison of various finite element methods with modern multigrid solvers, highlighting the efficiency of matrix-free implementations over traditional matrix-based approaches.

Findings

Matrix-free operator evaluation is up to ten times faster than sparse matrix-vector products.

Discontinuous Galerkin methods with matrix-free solvers achieve faster solutions than matrix-based hybridized DG.

Performance modeling confirms the advantages of matrix-free implementations for high-order finite element methods.

Abstract

This study presents a fair performance comparison of the continuous finite element method, the symmetric interior penalty discontinuous Galerkin method, and the hybridized discontinuous Galerkin method. Modern implementations of high-order methods with state-of-the-art multigrid solvers for the Poisson equation are considered, including fast matrix-free implementations with sum factorization on quadrilateral and hexahedral elements. For the hybridized discontinuous Galerkin method, a multigrid approach that combines a grid transfer from the trace space to the space of linear finite elements with algebraic multigrid on further levels is developed. Despite similar solver complexity of the matrix-based HDG solver and matrix-free geometric multigrid schemes with continuous and discontinuous Galerkin finite elements, the latter offer up to order of magnitude faster time to solution, even after including the superconvergence effects. This difference is because of vastly better performance of matrix-free operator evaluation as compared to sparse matrix-vector products. A roofline performance model confirms the advantage of the matrix-free implementation.