Diagonal Discrete Hodge Operators for Simplicial Meshes Using the Signed Dual Complex

We present a technique to extend the geometric construction of diagonal discrete Hodge operators to arbitrary triangular and tetrahedral boundary conforming Delaunay meshes in the frequent case of piecewise uniform and isotropic material parameters. The technique is based on the novel concept of signed dual complex that originates from a physical argument. In particular, it is shown how the positive definiteness of the mass matrix obtained with the signed dual complex is easily ensured for all boundary conforming Delaunay meshes without requiring-as expected by the common knowledge-that each circumcenter has to lie inside the corresponding element. Eliminating this requirement, whose fulfillment presents otherwise formidable practical difficulties, enables one to easily obtain efficient, consistent, and stable schemes.