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

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 (HDG) 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 HDG 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. It is found that high-order continuous finite elements give best time to solution for smooth solutions, closely followed by the matrix-free solvers for the other two discretizations. Their performance is up to an order of magnitude higher than that of the best matrix-based methods, even after including the superconvergence effects in the matrix-based HDG method. This difference is because of the vastly better performance of matrix-free operator evaluation as compared to sparse matrix-vector products. A roofline performance model confirms the superiority of the matrix-free implementation.