Additive Schwarz methods for serendipity elements

TL;DR

This paper develops additive Schwarz methods for serendipity finite elements, providing efficient solvers with condition numbers independent of polynomial degree and mesh size, confirmed by numerical experiments.

Contribution

It introduces additive Schwarz methods tailored for serendipity elements, extending existing theory and demonstrating computational efficiency and optimal conditioning.

Findings

Patch smoothers achieve degree-independent conditioning.

The two-grid method is optimal with mesh and degree independence.

Numerical experiments confirm theoretical efficiency improvements.

Abstract

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.