Discrete exterior calculus

Abstract

This thesis presents the beginnings of a theory of discrete exterior calculus (DEC). Our approach is to develop DEC using only discrete combinatorial and geometric operations on a simplicial complex and its geometric dual. The derivation of these may require that the objects on the discrete mesh, but not the mesh itself, are interpolated. Our theory includes not only discrete equivalents of differential forms, but also discrete vector fields and the operators acting on these objects. Definitions are given for discrete versions of all the usual operators of exterior calculus. The presence of forms and vector fields allows us to address their various interactions, which are important in applications. In many examples we find that the formulas derived from DEC are identical to the existing formulas in the literature. We also show that the circumcentric dual of a simplicial complex plays a useful role in the metric dependent part of this theory. The appearance of dual complexes leads to a proliferation of the operators in the discrete theory. One potential application of DEC is to variational problems which come equipped with a rich exterior calculus structure. On the discrete level, such structures will be enhanced by the availability of DEC. One of the objectives of this thesis is to fill this gap. There are many constraints in numerical algorithms that naturally involve differential forms. Preserving such features directly on the discrete level is another goal, overlapping with our goals for variational problems. In this thesis we have tried to push a purely discrete point of view as far as possible. We argue that this can only be pushed so far, and that interpolation is a useful device. For example, we found that interpolation of functions and vector fields is a very convenient. In future work we intend to continue this interpolation point of view, extending it to higher degree forms, especially in the context of the sharp, Lie derivative and interior product operators. Some preliminary ideas on this point of view are presented in the thesis. We also present some preliminary calculations of formulas on regular nonsimplicial complexes.