An ADI type operator splitting WG-FEM for 2D nonlinear unsteady singularly perturbed problem

Kumar N., Singh J., TOPRAKSEVEN Ş., Jiwari R.

Numerical Algorithms, 2025 (SCI-Expanded, Scopus) identifier

  • Yayın Türü: Makale / Tam Makale
  • Basım Tarihi: 2025
  • Doi Numarası: 10.1007/s11075-025-02061-5
  • Dergi Adı: Numerical Algorithms
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Applied Science & Technology Source, Computer & Applied Sciences, INSPEC, MathSciNet, zbMATH, DIALNET
  • Anahtar Kelimeler: ADI type scheme, Bakhvalov-type mesh, Crank-Nicolson scheme, Operator splitting, Parabolic semilinear equation, Singular perturbation, WG-FEM
  • Yozgat Bozok Üniversitesi Adresli: Evet

Özet

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.