  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.
