A Weak Galerkin Finite Element Method for the Maxwell Equations

• Published in: Journal of scientific computing Vol. 65; no. 1; pp. 363 - 386
• Main Authors: Mu, Lin, Wang, Junping, Ye, Xiu, Zhang, Shangyou
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.10.2015; Springer Nature B.V
• Subjects: Algorithms; Approximation; Boundary conditions; Computational Mathematics and Numerical Analysis; Continuity (mathematics); Convergence; Electric fields; Finite element analysis; Finite element method; Fourier transforms; Galerkin method; Linear equations; Magnetic fields; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Maxwell's equations; Methods; Numerical analysis; Operators (mathematics); Partial differential equations; Polynomials; Theoretical; Variables; Secondary: 35B45; 35J50; Weak gradient; Polyhedral meshes; Weak curl; Weak Galerkin; Finite element methods; Primary: 65N15; 76D07; Maxwell equations; 65N30
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-014-9964-4

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.