Implementing a Restricted Function Space Class in Firedrake

TL;DR

This paper details the development and implementation of a new RestrictedFunctionSpace class in Firedrake, enhancing solver capabilities by better handling Dirichlet boundary conditions and improving eigensolver performance.

Contribution

The paper introduces a novel RestrictedFunctionSpace class in Firedrake that addresses limitations in imposing boundary conditions and improves eigensolver accuracy.

Findings

Enhanced eigensolver performance with the new class

Effective removal of boundary-related eigenvalues

Demonstrated improvements through tests and comparisons

Abstract

The implementation process of a $\texttt{RestrictedFunctionSpace}$ class in Firedrake, a Python library which numerically solves partial differential equations through the use of the finite element method, is documented. This includes an introduction to the current $\texttt{FunctionSpace}$ class in Firedrake, and the key features that it has. With the current $\texttt{FunctionSpace}$ class, the limitations of the capabilities of the solvers in Firedrake when imposing Dirichlet boundary conditions are explored, as well as what the $\texttt{RestrictedFunctionSpace}$ class does differently to remove these issues. These will be considered in both a mathematical way, and in the code as an abstraction of the mathematical ideas presented. Finally, the benefits to the user of the $\texttt{RestrictedFunctionSpace}$ class are considered, and demonstrated through tests and comparisons. This leads to the conclusion that in particular, the eigensolver in Firedrake is improved through the use of the $\texttt{RestrictedFunctionSpace}$, through the removal of eigenvalues associated with the Dirichlet boundary conditions for a system.