Convergence analysis of a weak Galerkin finite element method on a Shishkin mesh for a singularly perturbed fourth-order problem in 2D

TL;DR

This paper analyzes the convergence of a weak Galerkin finite element method on Shishkin meshes for solving a challenging singularly perturbed fourth-order PDE in 2D, demonstrating robustness and accuracy.

Contribution

It introduces a convergence analysis for a weak Galerkin method on Shishkin meshes tailored for singularly perturbed fourth-order problems, with proofs and numerical validation.

Findings

Proves convergence in a discrete H^2 norm for the WG method.

Shows robustness of the method on Shishkin meshes with $N^2$ elements.

Provides numerical results confirming theoretical analysis.

Abstract

We consider the singularly perturbed fourth-order boundary value problem $\varepsilon ^{2}\Delta ^{2}u-\Delta u=f $ on the unit square $\Omega \subset \mathbb{R}^2$, with boundary conditions $u = \partial u / \partial n = 0$ on $\partial \Omega$, where $\varepsilon \in (0, 1)$ is a small parameter. The problem is solved numerically by means of a weak Galerkin(WG) finite element method, which is highly robust and flexible in the element construction by using discontinuous piecewise polynomials on finite element partitions consisting of polygons of arbitrary shape. The resulting WG finite element formulation is symmetric, positive definite, and parameter-free. Under reasonable assumptions on the structure of the boundary layers that appear in the solution, a family of suitable Shishkin meshes with $N^2$ elements is constructed ,convergence of the method is proved in a discrete $H^2$ norm for the corresponding WG finite element solutions and numerical results are presented.