The computational algorithm and numerical analysis of the signed diagonal Hodge star operator

Abstract

Differential form is an alternative mathematical form to describe the field variables and the operators in electromagnetism. From the viewpoint of differential forms, discretization of electromagnetic field is divided into two steps, i) discretization of the exterior derivative operator (Maxwell’s equations), and ii) discretization of the Hodge star operator (constitutive equations). The first step, the discrete form of the Maxwell’s equations based on differential forms has been obtained by other researchers. In contrast, the discrete Hodge star operators (discrete constitutive equations) have not been obtained so far. In the previous studies, unsigned diagonal discrete Hodge star operators are defined using the unsigned area and length for circumcenter dual meshes, however, it does not lead to correct solution of partial differential equations in the general Delaunay meshes. In this paper, we propose a definition of the signed diagonal discrete Hodge star using the signed area and length operator for circumcenter dual meshes. Also, based on this definition, we propose a simple practical calculation method for the signed discrete Hodge star operators. The result of convergence experiment indicates that the signed diagonal Hodge star operators produce the correct numerical solution for the general Delaunay meshes. Therefore, this definition and calculation method for the signed discrete Hodge star operator provides us with the explicit dynamics formulation for finite element analysis of electromagnetic field.