Operator-adapted wavelets for finite-element differential forms

Finite element differential forms

A differential form is a field which assigns to each point of a domain an alternating multilinear form on its tangent space. The exterior derivative operation, which maps differential forms to differential forms of the next higher order, unifies the basic first order differential operators of calculus, and is a building block for a great variety of differential equations. When discretizing such...

Finite element differential forms on cubical meshes

We develop a family of finite element spaces of differential forms defined on cubical meshes in any number of dimensions. The family contains elements of all polynomial degrees and all form degrees. In two dimensions, these include the serendipity finite elements and the rectangular BDM elements. In three dimensions they include a recent generalization of the serendipity spaces, and new H(curl)...

Finite Element Wavelets

Finite Element Wavelets on Manifolds

Preprint No. 1199, Department of Mathematics, University of Utrecht, June 2001. Submitted to IMA J. Numer. Anal. We construct locally supported, continuous wavelets on manifolds Γ that are given as the closure of a disjoint union of general smooth parametric images of an n-simplex. The wavelets are proven to generate Riesz bases for Sobolev spaces Hs(Γ) when s ∈ (−1, 3 2 ), if not limited by th...

Tensor Product Finite Element Differential Forms and Their Approximation Properties

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...