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

TL;DR

This paper introduces an efficient weak Galerkin finite element method with reduced stabilizers for Maxwell equations on complex polyhedral meshes, achieving optimal error estimates and demonstrating stability on both convex and non-convex geometries.

Contribution

It develops a stabilizer-reduced WG method capable of handling non-convex polyhedral meshes, improving efficiency and extending applicability beyond convex elements.

Findings

Achieves optimal error estimates in discrete $H^1$ and $L^2$ norms.

Demonstrates stability and efficiency through numerical experiments.

Extends WG methods to non-convex 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.