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.