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

Chunmei Wang, Shangyou Zhang

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.

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 norm, and additionally, an optimal error estimate is established for the WG solution. Several numerical experiments are conducted to validate the method's efficiency and accuracy.

Paper Structure

This paper contains 9 sections, 12 theorems, 138 equations, 4 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Weak Formulations
  3. Weak Differential Operators
  4. Weak Galerkin Algorithm
  5. Solution Existence and Uniqueness
  6. Error Equations
  7. Error Estimates
  8. $L^2$ Error Estimates
  9. Numerical test

Key Result

lemma 1

For ${\mathbf{v}}=\{{\mathbf{v}}_0, {\mathbf{v}}_b\}\in V_h$, there exists a constant $C$ such that

Figures (4)

  • Figure 1: The first three grids for the computation in Tables \ref{['t-1']}--\ref{['t-2']}.
  • Figure 2: The first three non-convex polygonal grids for the computation in Tables \ref{['t-3']}--\ref{['t-4']}.
  • Figure 3: The first two grids for the computation in Table \ref{['t-5']}.
  • Figure 4: The first three grids for the computation in Table \ref{['t-6']}.

Theorems & Definitions (25)

  • lemma 1
  • proof
  • Remark 5.1
  • lemma 2
  • proof
  • lemma 3
  • proof
  • Remark 5.2
  • theorem 1
  • proof
  • ...and 15 more