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This study presents a fair performance comparison of the continuous finiteelement method, the symmetric interior penalty discontinuous Galerkin method,and the hybridized discontinuous Galerkin method. Modern implementations ofhigh-order methods with state-of-the-art multigrid solvers for the Poissonequation are considered, including fast matrix-free implementations with sumfactorization on quadrilateral and hexahedral elements. For the hybridizeddiscontinuous Galerkin method, a multigrid approach that combines a gridtransfer from the trace space to the space of linear finite elements withalgebraic multigrid on further levels is developed. Despite similar solvercomplexity of the matrix-based HDG solver and matrix-free geometric multigridschemes with continuous and discontinuous Galerkin finite elements, the latteroffer up to order of magnitude faster time to solution, even after includingthe superconvergence effects. This difference is because of vastly betterperformance of matrix-free operator evaluation as compared to sparsematrix-vector products. A roofline performance model confirms the advantage ofthe matrix-free implementation.

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