Parallelized Discrete Exterior Calculus for Three-Dimensional Elliptic Problems

Abstract: A formulation of elliptic boundary value problems is used to develop the first discrete exterior calculus (DEC) library for massively parallel computations with 3D domains. This can be used for steady-state analysis of any physical process driven by the gradient of a scalar quantity, e.g. temperature, concentration, pressure or electric potential, and is easily extendable to transient analysis. In addition to offering this library to the community, we demonstrate one important benefit from the DEC formulation: effortless introduction of strong heterogeneities and discontinuities. These are typical for real materials, but challenging for widely used domain discretization schemes, such as finite elements. Specifically, we demonstrate the efficiency of the method for calculating the evolution of thermal conductivity of a solid with a growing crack population. Future development of the library will deal with transient problems, and more importantly with processes driven by gradients of vector quantities.