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We present an algorithmic framework for matrix-free evaluation ofdiscontinuous Galerkin finite element operators based on sum factorization onquadrilateral and hexahedral meshes. We identify a set of kernels for fastquadrature on cells and faces targeting a wide class of weak forms originatingfrom linear and nonlinear partial differential equations. Different algorithmsand data structures for the implementation of operator evaluation are comparedin an in-depth performance analysis. The sum factorization kernels areoptimized by vectorization over several cells and faces and an even-odddecomposition of the one-dimensional compute kernels. In isolation ourimplementation then reaches up to 60\% of arithmetic peak on Intel Haswell andBroadwell processors and up to 50\% of arithmetic peak on Intel KnightsLanding. The full operator evaluation reaches only about half that throughputdue to memory bandwidth limitations from loading the input and output vectors,MPI ghost exchange, as well as handling variable coefficients and the geometry.Our performance analysis shows that the results are often within 10\% of theavailable memory bandwidth for the proposed implementation, with the exceptionof the Cartesian mesh case where the cost of gather operations and MPIcommunication are more substantial.

Fast matrix-free evaluation of discontinuous Galerkin finite element operators