*Wissenschaftliches Programm* *Liste der Vortragenden*

**Sektion 6**

Donnerstag, 21.09.2000,
14.00–14.50 Uhr, WIL C 129

#### Ralf Hiptmair, Universität Tübingen, Sonderforschungsbereich 382

Many linear boundary value problems that occur in physical models can be stated in the calculus of
differential forms. In my talk I am trying to convey that this geometric perspective provides new insights
into the process of discretization and that we reap the possibility of a unified analysis of many different
discretization schemes.

The viewpoint of differential forms teaches us that one has to distinguish between topological equations
and metric-dependent constitutive laws. A straightforward discretization of the former is available
through using discrete differential forms. This results in generalized network equations that
completely preserve the topological features of the continuous problem also in the discrete
setting.

However, the constitutive laws defy a canonical treatment. Their formulation relies on the so-called
Hodge-operator, which lacks a clear discrete counterpart. I propose a few fundamental algebraic
requirements that have to be satisfied by meaningful discrete Hodge-operators, i.e. discrete material laws.
However general, they permit us to obtain a-priori error estimates.

It turns out that many discretization schemes ranging from primal and dual finite elements to finite
volume methods fit the framework and can be regarded as particular realizations of discrete
Hodge-operators.