An ADI type operator splitting WG-FEM for 2D nonlinear unsteady singularly perturbed problem
Numerical Algorithms, vol.101, no.3, pp.1701-1738, 2026 (SCI-Expanded, Scopus)
- Publication Type: Article / Article
- Volume: 101 Issue: 3
- Publication Date: 2026
- Doi Number: 10.1007/s11075-025-02061-5
- Journal Name: Numerical Algorithms
- Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Applied Science & Technology Source, Computer & Applied Sciences, INSPEC, MathSciNet, zbMATH, DIALNET
- Page Numbers: pp.1701-1738
- Keywords: ADI type scheme, Bakhvalov-type mesh, Crank-Nicolson scheme, Operator splitting, Parabolic semilinear equation, Singular perturbation, WG-FEM
- Yozgat Bozok University Affiliated: Yes
Abstract
In this study, we present an operator splitting alternate direction implicit (ADI) scheme via the weak Galerkin finite element method (WG-FEM) to effectively solve the two-dimensional semilinear parabolic singularly perturbed problem. First, we decompose the 2D model equation into two lower-dimensional 1D sub-problems using the operator splitting ADI method and couple these sub-problems through the initial conditions. Each 1D sub-problem is addressed using the WG-FEM in each spatial direction, combined with the Crank-Nicolson scheme for full discretization on a layer-adapted mesh. Additionally, in the error analysis section, we introduce an L2 projection as an intermediate operator for each spatial variable. Our main findings indicate that the proposed method achieves a convergence order of O(N-k) in each spatial direction and second-order convergence in the time direction. Finally, several numerical examples are performed to illustrate the effectiveness of the proposed method and confirm the theoretical findings.