  The approximation properties of the finite element method can often be
substantially improved by choosing smooth high-order basis functions. It is
extremely difficult to devise such basis functions for partitions consisting of
arbitrarily shaped polytopes. We propose the mollified basis functions of
arbitrary order and smoothness for partitions consisting of convex polytopes.
On each polytope an independent local polynomial approximant of arbitrary order
is assumed. The basis functions are defined as the convolutions of the local
approximants with a mollifier. The mollifier is chosen to be smooth, to have a
compact support and a unit volume. The approximation properties of the obtained
basis functions are governed by the local polynomial approximation order and
mollifier smoothness. The convolution integrals are evaluated numerically first
by computing the boolean intersection between the mollifier and the polytope
and then applying the divergence theorem to reduce the dimension of the
integrals. The support of a basis function is given as the Minkowski sum of the
respective polytope and the mollifier. The breakpoints of the basis functions,
i.e. locations with non-infinite smoothness, are not necessarily aligned with
polytope boundaries. Furthermore, the basis functions are not boundary
interpolating so that we apply boundary conditions with the non-symmetric
Nitsche method as in immersed/embedded finite elements. The presented numerical
examples confirm the optimal convergence of the proposed approximation scheme
for Poisson and elasticity problems.
