2023/06/28 by Shicheng Liu, Liu, Shicheng, Xiangyun Meng +3 · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Differential Equations and Numerical Methods #FOS: Mathematics #Material Science and Thermodynamics #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2306.15867
openalex publication_date 2023/06/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the singularly perturbed fourth-order boundary value problem ε 2Δ2u-Δu=f on the unit square Ω⊂ ℝ2, with boundary conditions u = ∂ u / ∂ n = 0 on ∂ Ω, where ε ∈ (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 N2 elements is constructed ,convergence of the method is proved in a discrete H2 norm for the corresponding WG finite element solutions and numerical results are presented.