A new mixed discontinuous Galerkin method for the electrostatic field

Abstract

We introduce and analyze a new mixed discontinuous Galerkin method for approximation of an electric field. We carry out its error analysis and prove an error estimate that is optimal in the mesh size. Some numerical results are given to confirm the theoretical convergence.

An Augmented Mixed DG Scheme for the Electric Field

“…The primal DG method consists to replace the traces of functions used in (2.1)-(2.2) by numerical fluxes, this is due to the discontinuity of solution (u, p) on interfaces of the triangulation. For the definitions of averages and jumps of a discontinuous function on the interfaces we refer to [16]. As in [1], the numerical fluxes are chosen to provide a numerical scheme consistent and conservative.…”

“…In the last years, the Maxwell equations have been studied and analysed by using several numerical methods such as discontinuous Galerkin methods [5,6,9,10,[13][14][15][16] and by weak Galerkin formulations [18]. Thanks to works of Cockburn et al [1][2][3], DG methods are developed very well and it was applied to solve numerically many problems of partial differential equations like Poisson's equation [2], Stokes equations [7] , Maxwell's equations [5,6,8,10,13,17].…”

“…The researchers C. Daveau and A. Zaghdani study Maxwell equations and wave equation in [4][5][6] by using some new schemes of DG methods. In [16,17], A. Zaghdani et al presented a mixed DG scheme for the numerical resolution of the electrostatic field. This work is an expansion of [5] where the problem (1.1) was considered with constant permeability and permittivity coefficients and when α = 0, the problem (1.1) was also analysed in [9], however the formulation of equations exploited to find the error estimates is not consistent, this is due to the choice of the lifting operators.…”

Analysis of aWeak Galerkin Mixed Formulation for Maxwell’s Equations

“…The idea of the weak Galerkin finite element method introduced by [13] consists in the approximation of the differential operators in the partial differential equation by weak forms as distributions over the space of discontinuous functions including boundary information. Compared to the discontinuous Galerkin methods [11,[16][17][18][19], the weak Galerkin methods also use discontinuous functions in the finite element procedure which gives a great flexibility to the WG-FEM in dealing with boundary conditions and different geometric complexities, while weak Galerkin methods require only weak continuity of variables through well-defined discrete differential operators and are absolutely stable when correctly constructed. Ever since it was introduced, the WG-method was used by several authors for the resolution of various partial differential equations such as linear parabolic problems [3,20,21], Helmoltz equations with large wave numbers [12] and elliptic interface problems [7,8].…”