A Modified weak Galerkin finite element method for the Maxwell equations on polyhedral meshes

TL;DR

This paper presents a modified weak Galerkin finite element method for solving time-harmonic Maxwell's equations on polyhedral meshes, achieving optimal convergence and improved effectiveness over existing methods.

Contribution

The paper introduces a novel modified weak Galerkin approach that uses averaged inter-element functions, enabling direct definition of weak curl and gradient on discontinuous polynomials.

Findings

Optimal-order convergence proved theoretically.

Numerical examples confirm the method's effectiveness.

Outperforms existing weak Galerkin methods.

Abstract

We introduce a new numerical method for solving time-harmonic Maxwell's equations via the modified weak Galerkin technique. The inter-element functions of the weak Galerkin finite elements are replaced by the average of the two discontinuous polynomial functions on the two sides of the polygon, in the modified weak Galerkin (MWG) finite element method. With the dependent inter-element functions, the weak curl and the weak gradient are defined directly on totally discontinuous polynomials. Optimal-order convergence of the method is proved. Numerical examples confirm the theory and show effectiveness of the modified weak Galerkin method over the existing methods.