  Unfitted finite element methods, like CutFEM, have traditionally been
implemented in a matrix-based fashion, where a sparse matrix is assembled and
later applied to vectors while solving the resulting linear system. With the
goal of increasing performance and enabling algorithms with polynomial spaces
of higher degrees, this contribution chooses a more abstract approach by
matrix-free evaluation of the operator action on vectors instead. The proposed
method loops over cells and locally evaluates the cell, face, and interface
integrals, including the contributions from cut cells and the different means
of stabilization. The main challenge is the efficient numerical evaluation of
terms in the weak form with unstructured quadrature points arising from the
unfitted discretization in cells cut by the interface. We present design
choices and performance optimizations for tensor-product elements and
demonstrate the performance by means of benchmarks and application examples. We
demonstrate a speedup of more than one order of magnitude for the operator
evaluation of a discontinuous Galerkin discretization with polynomial degree
three compared to a sparse matrix-vector product and develop performance models
to quantify the performance properties over a wide range of polynomial degrees.
