4 research outputs found
Spaces of finite element differential forms
We discuss the construction of finite element spaces of differential forms
which satisfy the crucial assumptions of the finite element exterior calculus,
namely that they can be assembled into subcomplexes of the de Rham complex
which admit commuting projections. We present two families of spaces in the
case of simplicial meshes, and two other families in the case of cubical
meshes. We make use of the exterior calculus and the Koszul complex to define
and understand the spaces. These tools allow us to treat a wide variety of
situations, which are often treated separately, in a unified fashion.Comment: To appear in: Analysis and Numerics of Partial Differential
Equations, U. Gianazza, F. Brezzi, P. Colli Franzone, and G. Gilardi, eds.,
Springer 2013. v2: a few minor typos corrected. v3: a few more typo
correction
Constructions of some minimal finite element systems
Within the framework of finite element systems, we show how spaces of
differential forms may be constructed, in such a way that they are equipped
with commuting interpolators and contain prescribed functions, and are minimal
under these constraints. We show how various known mixed finite element spaces
fulfill such a design principle, including trimmed polynomial differential
forms, serendipity elements and TNT elements. We also comment on virtual
element methods and provide a dimension formula for minimal compatible finite
element systems containing polynomials of a given degree on hypercubes.Comment: Various minor changes, based on suggestions of paper referee