We define signed dual volumes at all dimensions for circumcentric dual meshes. We show that for pairwise Delaunay triangulations with mild boundary assumptions these signed dual volumes are positive. This allows the use of such Delaunay meshes for Di
screte Exterior Calculus (DEC) because the discrete Hodge star operator can now be correctly defined for such meshes. This operator is crucial for DEC and is a diagonal matrix with the ratio of primal and dual volumes along the diagonal. A correct definition requires that all entries be positive. DEC is a framework for numerically solving differential equations on meshes and for geometry processing tasks and has had considerable impact in computer graphics and scientific computing. Our result allows the use of DEC with a much larger class of meshes than was previously considered possible.
There are very few results on mixed finite element methods on surfaces. A theory for the study of such methods was given recently by Holst and Stern, using a variational crimes framework in the context of finite element exterior calculus. However, we
are not aware of any numerical experiments where mixed finite elements derived from discretizations of exterior calculus are used for a surface domain. This short note shows results of our preliminary experiments using mixed methods for Darcy flow (hence scalar Poissons equation in mixed form) on surfaces. We demonstrate two numerical methods. One is derived from the primal-dual Discrete Exterior Calculus and the other from lowest order finite element exterior calculus. The programming was done in the language Python, using the PyDEC package which makes the code very short and easy to read. The qualitative convergence studies seem to be promising.
We describe algorithms for finding harmonic cochains, an essential ingredient for solving elliptic partial differential equations in exterior calculus. Harmonic cochains are also useful in computational topology and computer graphics. We focus on fin
ding harmonic cochains cohomologous to a given cocycle. Amongst other things, this allows localization near topological features of interest. We derive a weighted least squares method by proving a discrete Hodge-deRham theorem on the isomorphism between the space of harmonic cochains and cohomology. The solution obtained either satisfies the Whitney form finite element exterior calculus equations or the discrete exterior calculus equations for harmonic cochains, depending on the discrete Hodge star used.
