Discrete Differential Forms on Simplices

R. Hiptmair

Sonderforschungsbereich 382 Universität Tübingen

e–mail: hiptmair@na.uni-tuebingen.de URL: http://na.uni-tuebingen.de/˜ hiptmair

We are faced with a wealth of mixed finite element schemes that supply vector-valued finite elements conforming in either H(curl, Ω) or H(div, Ω). Important specimens are the Raviart-Thomas family of finite elements, widely used for the mixed discretization of second order elliptic problems, or the so-called edge elements, which have become hugely popular in computational electromagnetism. These spaces were introduced independently, and with much technical effort in each case.

At first glance, the polynomial local spaces for all these schemes seem to have little in common. The same is true of the definitions of degrees of freedom, in particular in the higher order cases. What are the hidden links? In the case of simplicial meshes in any dimension, I take a fresh look at this issue heavily relying on the calculus of differential forms: Finite element functions are regarded as discrete differential forms. This makes it possible to exploit the close relationships between forms of different orders and forms in different dimensions.

Thus, it is possible to come up with a single representation formula describing the local spaces for all simplicial discrete differential forms. The dimension of the ambient space, the order of the form, and its polynomial degree enter as parameters. The crucial device for doing this is the Poincaré mapping, normally used to construct potentials for closed forms.

The degrees of freedoms allow a unifying description, too. They can be stated as integrals over sub-simplices that involve exterior products with appropriate “weight forms”. Unisolvence can be shown, as well as the capacity of these degrees of freedom to ensure the necessary global continuity conditions. Obvious from the construction is the fundamental commuting sequence property.

At one fell swoop the exact sequence property can be established for the families of discrete differential forms. We also learn that hierarchical bases provide local discrete potentials of higher polynomial order.

References

[1] A. Bossavit, Mixed finite elements and the complex of Whitney forms, in The Mathematics of Finite Elements and Applications VI, J. Whiteman, ed., Academic Press, London, 1988, pp. 137–144.

[2] R. Hiptmair, Canonical construction of finite elements, Math. Comp., 68 (1999), pp. 1325–1346.

[3] , Higher order Whitney forms, Progress in Electromagnetics Research, PIER, 32 (2001), pp. 271– 299. Abstract in Journal of Electromagnetic Waves and Applications, 15(3), 341-342, 2001.