Tensor product finite element differential forms and their approximation properties

01/2013

Tensor product finite element differential forms and their approximation properties

Friday 4th January 2013

Arnold, D.N.; Boffi, D.; Bonizzoni,F.

We discuss the tensor product construction for complexes of differential forms and show how it can be applied to define shape functions and degrees of freedom for finite element differential forms on cubes in n dimensions. These may be extended to curvilinear cubic elements, obtained as images of a reference cube under diffeomorphisms, by using the pullback transformation for differential forms to map the shape functions and degrees of freedom from the reference cube to the image finite element. This construction recovers and unifies several known finite element approximations in two and three dimensions. In this context, we study the approximation properties of the resulting finite element spaces in two particular cases: when the maps from the reference cube are affine, and when they are multilinear. In the former case the rate of convergence depends only on the degree of polynomials contained in the reference space of shape functions. In the latter case, the rate of approximation is degraded, with the loss more severe for differential forms of higher form degree.