Finite element differential forms on curvilinear cubic meshes and their approximation properties

TL;DR

This paper investigates the approximation properties of finite element differential forms on curvilinear cubic meshes, providing conditions for convergence rates based on mesh and shape function characteristics.

Contribution

It establishes a sufficient condition on reference shape functions to ensure desired convergence rates on curvilinear meshes.

Findings

Affine diffeomorphisms yield standard convergence rates.

Multilinear diffeomorphisms can degrade convergence, especially for higher form degrees.

Provided conditions help maintain convergence rates on curved meshes.

Abstract

We study the approximation properties of a wide class of finite element differential forms on curvilinear cubic meshes in n dimensions. Specifically, we consider meshes in which each element is the image of a cubical reference element under a diffeomorphism, and finite element spaces in which the shape functions and degrees of freedom are obtained from the reference element by pullback of differential forms. In the case where the diffeomorphisms from the reference element are all affine, i.e., mesh consists of parallelotopes, it is standard that the rate of convergence in L2 exceeds by one the degree of the largest full polynomial space contained in the reference space of shape functions. When the diffeomorphism is multilinear, the rate of convergence for the same space of reference shape function may degrade severely, the more so when the form degree is larger. The main result of the paper gives a sufficient condition on the reference shape functions to obtain a given rate of convergence.