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The stabilizer free weak Galerkin mixed finite elements method for the biharmonic equation

Shanshan Gu, Fuchang Huo, Shicheng Liu

Abstract

In this article, the stabilizer free weak Galerkin (SFWG) finite element method is applied to the Ciarlet-Raviart mixed form of the Biharmonic equation. We utilize the SFWG solutions of the second elliptic problems to define projection operators, build error equations, and further derive the error estimates. Finally, numerical examples support the results reached by the theory.

The stabilizer free weak Galerkin mixed finite elements method for the biharmonic equation

Abstract

In this article, the stabilizer free weak Galerkin (SFWG) finite element method is applied to the Ciarlet-Raviart mixed form of the Biharmonic equation. We utilize the SFWG solutions of the second elliptic problems to define projection operators, build error equations, and further derive the error estimates. Finally, numerical examples support the results reached by the theory.
Paper Structure (10 sections, 19 theorems, 100 equations, 3 figures, 8 tables, 1 algorithm)

This paper contains 10 sections, 19 theorems, 100 equations, 3 figures, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. SFWG mixed scheme for the Biharmonic equation
  3. Well-posedness
  4. Error analysis
  5. Ritz and Neumann projections
  6. Error equations
  7. Error estimates
  8. Numerical Results
  9. Conclusion
  10. Some Technical Results of Ritz and Neumann Projections

Key Result

Lemma 3.1

PossionSFWGThere exist two positive constants $C_1$ and $C_2$ such that where $j=n+k-1$ ($n$ is the number of edges of the polygon) in the definition of $\nabla _w$.

Figures (3)

  • Figure 1: The uniform triangular meshes with $n=2,~4,~8$
  • Figure 2: The uniform rectangular meshes with $n=2,~4,~8$
  • Figure 3: The polygon meshes with $n=2,~4,~8$

Theorems & Definitions (44)

  • Definition 2.1
  • Definition 3.1
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • ...and 34 more