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 work tackles the singularly perturbed 2D fourth-order boundary value problem $\varepsilon^2\Delta^2 u-\Delta u=f$ on $\Omega=(0,1)^2$ with clamped boundary conditions, using a weak Galerkin finite element method on a Shishkin mesh. A discrete weak Laplacian $\Delta_w$ and weak gradient $\nabla_w$ are constructed, along with a stabilizer, to form a symmetric, positive definite, parameter-free scheme that is solved in a broken finite element space. The authors prove an a priori error bound in a discrete $H^2$-equivalent norm that is essentially independent of $\varepsilon$, achieving an order $N^{-(k-1)}$ with logarithmic factors for $u\in H^{k+1}(\Omega)$, and verify the theory with numerical experiments on Shishkin meshes. The results demonstrate ε-independent convergence and improved accuracy on Shishkin meshes compared to uniform meshes, confirming the practical robustness of the WG approach for singular perturbations in 2D.

Abstract

We consider the singularly perturbed fourth-order boundary value problem $\varepsilon ^{2}Δ^{2}u-Δu=f $ on the unit square $Ω\subset \mathbb{R}^2$, with boundary conditions $u = \partial u / \partial n = 0$ on $\partial Ω$, 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.

Figures (1)

• Figure 1: A rectangular Shishkin mesh with $N = 8$ and dissection of $\Omega$.

Theorems & Definitions (17)

• Lemma 3.1
• proof
• Lemma 4.1
• proof
• Lemma 4.2
• proof
• Lemma 4.3
• Lemma 4.4
• Lemma 4.5
• proof