Discrete differential forms
Ralf Hiptmair
Institut fur Angewandte Mathematik
Universitat Bonn, Germany
The most natural mathematical model for electromagnetic fields is supplied
by the
calculus of differential forms, which can be understood as mappings
assigning real
numbers to oriented bounded sub-manifolds of the affine space
R3. In
this talk I am trying to convey that suitable finite elements on simplicial
triangulations, most notably the so-called edge elements, emerge as discrete
differential forms in a straightforward fashion. In particular, in the case
of lowest
polynomial degree they can be obtained by a natural interpolation procedure. The
commuting diagram property of nodal interpolation and the existence of discrete
potentials is immediate.
The situation is more complicated in the case of higher polynomial degree.
In this
case local Helmholtz-type decompositions of spaces of polynomial
differential forms
are the key to the construction of discrete differential forms. A unifying
description of degrees of freedom as moments over mesh facets can be given.
We also
learn that hierarchical bases provide local discrete potentials of higher
polynomial order.
[1] R. Hiptmair, Canonical construction of finite elements.
Math. Comp. 68 (1999) 1325-1346.
[2] R. Hiptmair, Finite elements in computational electromagnetism. Acta
Numerica (2002) 237-339.