  While solving Partial Differential Equations (PDEs) with finite element
methods (FEM), serendipity elements allow us to obtain the same order of
accuracy as rectangular tensor-product elements with many fewer degrees of
freedom (DOFs). To realize the possible computational savings, we develop some
additive Schwarz methods (ASM) based on solving local patch problems. Adapting
arguments from Pavarino for the tensor-product case, we prove that patch
smoothers give conditioning estimates independent of the polynomial degree for
a model problem. We also combine this with a low-order global operator to give
an optimal two-grid method, with conditioning estimates independent of the mesh
size and polynomial degree. The theory holds for serendipity elements in two
and three dimensions, and can be extended to full multigrid algorithms.
Numerical experiments using Firedrake and PETSc confirm this theory and
demonstrate efficiency relative to standard elements.
