Local coderivatives and approximation of Hodge Laplace problems

TL;DR

This paper develops local finite element methods for Hodge Laplace problems that balance finite element choices and numerical integration to improve locality of coderivative approximations, supported by convergence analysis.

Contribution

It introduces new local methods for Hodge Laplace problems that address nonlocal coderivative approximations in mixed finite element schemes.

Findings

Constructed local finite element schemes for Hodge Laplace problems.

Established convergence estimates using variational crime framework.

Demonstrated improved locality properties of coderivative approximations.

Abstract

The standard mixed finite element approximations of Hodge Laplace problems associated with the de Rham complex are based on proper discrete subcomplexes. As a consequence, the exterior derivatives, which are local operators, are computed exactly. However, the approximations of the associated coderivatives are nonlocal. In fact, this nonlocal property is an inherent consequence of the mixed formulation of these methods, and can be argued to be an undesired effect of these schemes. As a consequence, it has been argued, at least in special settings, that more local methods may have improved properties. In the present paper, we construct such methods by relying on a careful balance between the choice of finite element spaces, degrees of freedom, and numerical integration rules. Furthermore, we establish key convergence estimates based on a standard approach of variational crimes.