Finite element differential forms on curvilinear cubic meshes and their approximation properties
For researchers in numerical analysis and finite element methods, this work addresses a known bottleneck in convergence theory for curvilinear meshes.
The paper studies approximation properties of finite element differential forms on curvilinear cubic meshes, showing that convergence rates degrade for multilinear diffeomorphisms, and provides a sufficient condition on reference shape functions to achieve a given rate.
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.