  We present an algorithmic framework for matrix-free evaluation of
discontinuous Galerkin finite element operators based on sum factorization on
quadrilateral and hexahedral meshes. We identify a set of kernels for fast
quadrature on cells and faces targeting a wide class of weak forms originating
from linear and nonlinear partial differential equations. Different algorithms
and data structures for the implementation of operator evaluation are compared
in an in-depth performance analysis. The sum factorization kernels are
optimized by vectorization over several cells and faces and an even-odd
decomposition of the one-dimensional compute kernels. In isolation our
implementation then reaches up to 60\% of arithmetic peak on Intel Haswell and
Broadwell processors and up to 50\% of arithmetic peak on Intel Knights
Landing. The full operator evaluation reaches only about half that throughput
due 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 the
available memory bandwidth for the proposed implementation, with the exception
of the Cartesian mesh case where the cost of gather operations and MPI
communication are more substantial.
