Efficient Weak Galerkin Finite Element Methods for Maxwell Equations on polyhedral Meshes without Convexity Constraints

Abstract: This paper presents an efficient weak Galerkin (WG) finite element method with reduced stabilizers for solving the time-harmonic Maxwell equations on both convex and non-convex polyhedral meshes. By employing bubble functions as a critical analytical tool, the proposed method enhances efficiency by partially eliminating the stabilizers traditionally used in WG methods. This streamlined WG method demonstrates stability and effectiveness on convex and non-convex polyhedral meshes, representing a significant improvement over existing stabilizer-free WG methods, which are typically limited to convex elements within finite element partitions. The method achieves an optimal error estimate for the exact solution in a discrete $H^1$ norm, and additionally, an optimal $L^2$ error estimate is established for the WG solution. Several numerical experiments are conducted to validate the method's efficiency and accuracy.