Discrete Helmholtz–Hodge Decomposition on Polyhedral Meshes Using Compatible Discrete Operators

Abstract

This article provides a methodology to perform discrete Helmholtz---Hodge decomposition on three-dimensional polyhedral meshes using structure-preserving schemes: the Compatible Discrete Operator schemes. We propose to extract the decomposition components independently with one equation to solve per component or potential. The key of the method is the choice of a discrete Hodge operator that makes a compromise between convergence rate and computational cost. Numerical experiments are performed to evaluate the convergence rate and the computational cost on various polyhedral meshes, in particular, on the FVCA benchmark meshes. We also investigate some linear solver capabilities to solve our equations. The main contribution of this paper is the application of the CDO schemes to the Hodge decomposition and the required solvers.