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

TL;DR

This paper analyzes the anisotropic error behavior of the Weak Galerkin finite element method applied to singularly perturbed biharmonic problems, demonstrating uniform convergence and validating results through numerical examples.

Contribution

It provides the first anisotropic error analysis for Weak Galerkin methods on singularly perturbed biharmonic problems using Shishkin meshes.

Findings

Error estimates in $H^{2}$ norm established

Uniform convergence proved

Numerical results confirm theoretical analysis

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.