A Weak Galerkin Finite Element Method for the Maxwell Equations

Abstract

This paper introduces a numerical scheme for the time-harmonic Maxwell equations by using weak Galerkin (WG) finite element methods. The WG finite element method is based on two operators: discrete weak curl and discrete weak gradient, with appropriately defined stabilizations that enforce a weak continuity of the approximating functions. This WG method is highly flexible by allowing the use of discontinuous approximating functions on arbitrary shape of polyhedra and, at the same time, is parameter free. Optimal-order of convergence is established for the WG approximations in various discrete norms which are either H 1 -like or L 2 and L 2 -like. An effective implementation of the WG method is developed through variable reduction by following a Schur-complement approach, yielding a system of linear equations involving unknowns associated with element boundaries only. Numerical results are presented to confirm the theory of convergence.

Theoretical and numerical analysis of a non-local dispersion model for light interaction with metallic nanostructures

“…Hence, the existence and uniqueness of the solutionÊ ∈ H 0 (curl; Ω) andĴ ∈ H 0 (div; Ω) for (13)- (14) is guaranteed by the Lax-Milgram lemma. The existence and uniqueness ofĤ in H(curl; Ω) follows from (7)- (8).…”

“…Li et al [13] proposed an energy-conserved splitting type FDTD scheme for Maxwell's equations in metamaterials. Mu et al [14] extended the weak Galerkin method to solve the time-harmonic Maxwell's equations. Zhang et al [15] investigated both FDTD and DGTD methods for solving twodimensional Maxwell Interface problems.…”

A P0–P0 weak Galerkin finite element method for solving singularly perturbed reaction–diffusion problems

“…The WGFE technique has been rapidly utilized to different types of PDEs, including second order-elliptic equations [1][2][3][4][5][6][7][8][9], the biharmonic problems [10][11][12], the stokes problems [13], the Maxwell equations [14], the linear hyperbolic equations [15], and the Helmholtz equations [16]. The purpose of a WGFE method is to replace the partial differential operators by the corresponding generalized distributions, namely weak differential operators, on the space of discontinuous functions.…”

The Cascadic Multigrid Method of the Weak Galerkin Method for Second‐Order Elliptic Equation

“…In [4], the weak Galerkin mixed finite element method for biharmonic equations has been developed. For the applications of the weak Galerkin finite element method for other types of partial differential equations, the readers are referred to [5][6][7][8].…”