Discretization of div-curl Systems by Weak Galerkin Finite Element Methods on Polyhedral Partitions

TL;DR

This paper introduces a novel weak Galerkin finite element method for discretizing div-curl systems on polyhedral partitions, effectively handling heterogeneous media and different boundary conditions with proven optimal error estimates.

Contribution

The paper develops a new weak Galerkin discretization scheme for div-curl systems that handles heterogeneous media and boundary conditions without computing harmonic functions.

Findings

Achieved optimal error estimates in various norms.

Successfully discretized div-curl systems on polyhedral partitions.

Handled both normal and tangential boundary conditions effectively.

Abstract

In this paper, the authors devise a new discretization scheme for div-curl systems defined in connected domains with heterogeneous media by using the weak Galerkin finite element method. Two types of boundary value problems are considered in the algorithm development: (1) normal boundary condition, and (2) tangential boundary condition. A new variational formulation is developed for the normal boundary value problem by using the Helmholtz decomposition which avoids the computation of functions in the harmonic fields. Both boundary value problems are reduced to a general saddle-point problem involving the curl and divergence operators, for which the weak Galerkin finite element method is devised and analyzed. The novelty of the technique lies in the discretization of the divergence operator applied to vector fields with heterogeneous media. Error estimates of optimal order are established for the corresponding finite element approximations in various discrete Sobolev norms.