Bringing Trimmed Serendipity Methods to Computational Practice in Firedrake

TL;DR

This paper implements trimmed serendipity finite element methods in Firedrake, demonstrating their efficiency and comparable convergence to traditional elements for various PDE problems.

Contribution

It introduces an implementation of trimmed serendipity elements in Firedrake, enabling their use in practical computational problems involving $H^1$, \\hcurl, or \\hdiv-conforming elements.

Findings

Trimmed serendipity elements converge at the same rate as tensor product elements.

They offer significant savings in computational time or memory.

Effective for problems like Poisson and Maxwell eigenvalue problems.

Abstract

We present an implementation of the trimmed serendipity finite element family, using the open source finite element package Firedrake. The new elements can be used seamlessly within the software suite for problems requiring $H^1$, \hcurl, or \hdiv-conforming elements on meshes of squares or cubes. To test how well trimmed serendipity elements perform in comparison to traditional tensor product elements, we perform a sequence of numerical experiments including the primal Poisson, mixed Poisson, and Maxwell cavity eigenvalue problems. Overall, we find that the trimmed serendipity elements converge, as expected, at the same rate as the respective tensor product elements while being able to offer significant savings in the time or memory required to solve certain problems.