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Anisotropic Error Analysis of Weak Galerkin finite element method for Singularly Perturbed Biharmonic Problems

Aayushman Raina, Srinivasan Natesan, Şuayip Toprakseven

Abstract

We consider the Weak Galerkin finite element approximation of the Singularly Perturbed Biharmonic elliptic problem on a unit square domain with clamped boundary conditions. Shishkin mesh is used for domain discretization as the solution exhibits boundary layers near the domain boundary. Error estimates in the equivalent $H^{2}-$ norm have been established and the uniform convergence of the proposed method has been proved. Numerical examples are presented corroborating our theoretical findings.

Anisotropic Error Analysis of Weak Galerkin finite element method for Singularly Perturbed Biharmonic Problems

Abstract

We consider the Weak Galerkin finite element approximation of the Singularly Perturbed Biharmonic elliptic problem on a unit square domain with clamped boundary conditions. Shishkin mesh is used for domain discretization as the solution exhibits boundary layers near the domain boundary. Error estimates in the equivalent norm have been established and the uniform convergence of the proposed method has been proved. Numerical examples are presented corroborating our theoretical findings.
Paper Structure (24 sections, 13 theorems, 116 equations, 4 figures, 2 tables)

This paper contains 24 sections, 13 theorems, 116 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The model problem
  3. Solution decomposition
  4. The Shishkin triangular mesh
  5. Weak formulation and well-posedness
  6. A weak Laplacian and a weak gradient operator and its approximation
  7. Proof
  8. Proof
  9. Proof
  10. Proof
  11. Local discretization of weak Galerkin FEM
  12. Computation of local stiffness matrices
  13. Local discrete stabilizer term
  14. Error Analysis
  15. Interpolation operator
  16. ...and 9 more sections

Key Result

Lemma 2.1

The solution $u$ of the problem ch1_md_problem in a unit square has the following decomposition: where $K := \{1,2,3,4,21,41,23,43\}$, $\mathscr{S}$ is the smooth term and $\mathscr{E}_{i}$ are the layer terms with $\mathscr{S}, \mathscr{E}_{i} \in H^{s}(\Omega), i \in K$ for any natural number $s$ that satisfies $3 \leq s \leq k+1$. One can refer Figure ch1_mesh for better understanding. Furth w

Figures (4)

  • Figure 1: Discretization of unit square $\Bar{\Omega}$
  • Figure 2: A Shishkin triangular mesh with $N=8$
  • Figure 3: Comparison of solution for Example \ref{['ex1']} for $\varepsilon^2 = 10^{-6}$ and $N=32$.
  • Figure 4: Comparison of solution for Example \ref{['ex2']} for $\varepsilon^2 = 10^{-6}$ and $N=32$.

Theorems & Definitions (18)

  • Lemma 2.1: franz2014c0
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 3.1
  • Remark 4.1
  • Theorem 4.2: local
  • Lemma 4.3
  • Lemma 4.4
  • ...and 8 more