Core Concepts
The paper presents a convergence analysis of a weak Galerkin finite element method on a Shishkin mesh for solving a singularly perturbed fourth-order boundary value problem in 2D. The method exhibits uniform convergence regardless of the singular perturbation parameter.
Abstract
The paper focuses on solving a singularly perturbed fourth-order boundary value problem in a 2D domain using the weak Galerkin (WG) finite element method with a Shishkin mesh. The key highlights are:
The WG method is employed to solve the problem, which establishes distinct basis functions for the interior and boundary of each partitioned element, and uses a discretized weak differential operator instead of the traditional differential operator.
A Shishkin mesh is used to ensure uniform convergence of the method, regardless of the singular perturbation parameter ε. The mesh is refined in the layers to capture the layer structure in the solution.
An asymptotically optimal order error estimate in a H2-equivalent discrete norm is established for the corresponding WG solutions.
Numerical experiments are presented to verify the theoretical convergence results. The results show that the WG method on the Shishkin mesh achieves better performance compared to the uniform mesh, and the asymptotically optimal order of convergence is obtained.
Stats
ε2∆2u - ∆u = f, in Ω
u = 0, ∂u/∂n = 0, on ∂Ω
0 < ε ≪ 1, f ∈ L2(Ω)