A geometric description of Discrete Exterior Calculus for general triangulations

Abstract

We revisit the theory of Discrete Exterior Calculus (DEC) in 2D for general triangulations, relying only on Vector Calculus and Matrix Algebra. We present DEC numerical solutions of the Poisson equation and compare them against those found using the Finite Element Method with linear elements (FEML). Contents 1. Introduction 1 2. 2D Exterior Differential Calculus as Vector Calculus 2 2.1. Wedge product for vectors in R 2 2 2.2. Hodge star operator for vectors in R 2 4 2.3. The Laplacian 4 2.4. Duality in Green's theorem 5 3. Discrete Exterior Calculus 6 3.1. Boundary operator 6 3.2. Dual mesh 9 3.3. Boundary operator on the dual mesh 10 3.4. Discrete Hodge star 11 3.5. DEC applied to 2D Poisson's equation 13 4. DEC for general triangulations 13 4.1. Dual mesh of an arbitrary triangle 13 4.2. Dual mesh of a general triangulation 14 5. Numerical examples 15 5.1. First example 15 5.2. Second example 17 6. Conclusions 19 References 19

GPU-accelerated time integration of Gross-Pitaevskii equation with discrete exterior calculus

“…The implementation shows up to 152-fold performance improvement when benchmarked against a reference CPU version, and good scalability over the number of GPUs. There is no previous research on the topic of using GPUs in DEC. Esqueda et al [28] mention the usage of a single GPU in their numerical experiments, but give no description of the implementation nor its computing time performance, therefore this paper aims to fill those gaps.…”