High order approximation of Hodge Laplace problems with local coderivatives on cubical meshes

TL;DR

This paper extends high-order mixed finite element methods for Hodge Laplace problems on cubical meshes, ensuring local coderivatives through a new family of finite element differential forms, improving locality and computational properties.

Contribution

It introduces a new family of finite element differential forms on cubical meshes for high-order approximations, generalizing the linear degree and proving unisolvency.

Findings

Successfully extended low order methods to high orders on cubical meshes.

Established theoretical foundations for local coderivatives in high-order methods.

Enhanced the accuracy and locality of finite element approximations.

Abstract

In mixed finite element approximations of Hodge Laplace problems associated with the de Rham complex, the exterior derivative operators are computed exactly, so the spatial locality is preserved. However, the numerical approximations of the associated coderivatives are nonlocal and it can be regarded as an undesired effect of standard mixed methods. For numerical methods with local coderivatives a perturbation of low order mixed methods in the sense of variational crimes has been developed for simplicial and cubical meshes. In this paper we extend the low order method to all high orders on cubical meshes using a new family of finite element differential forms on cubical meshes. The key theoretical contribution is a generalization of the linear degree, in the construction of the serendipity family of differential forms, and the generalization is essential in the unisolvency proof of the new family of finite element differential forms.