WG-FEM With ADI-Type Splitting: A High-Order Approach for Two-Parameter Singularly Perturbed 2D Parabolic Problems on Bakhvalov Mesh

Kumar N., Toprakseven S., Singh J.

Numerical Methods for Partial Differential Equations, cilt.41, sa.5, 2025 (SCI-Expanded, Scopus) identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 41 Sayı: 5
  • Basım Tarihi: 2025
  • Doi Numarası: 10.1002/num.70025
  • Dergi Adı: Numerical Methods for Partial Differential Equations
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Aerospace Database, Communication Abstracts, Compendex, MathSciNet, zbMATH, DIALNET, Civil Engineering Abstracts
  • Anahtar Kelimeler: ADI scheme, Bakhvalov-type mesh, balanced norm, parabolic reaction-convection-diffusion, singular perturbation, two parameters, weak Galerkin method
  • Yozgat Bozok Üniversitesi Adresli: Evet

Özet

This study presents a novel computational approach that integrates the operator splitting alternate direction implicit (ADI) scheme with a high-order weak Galerkin finite element method to solve a two-dimensional singularly perturbed parabolic reaction-convection-diffusion problem involving two small parameters. The ADI technique decomposes the original 2D problem into a sequence of 1D subproblems along each spatial direction, which are then discretized using the weak Galerkin finite element method on a nonuniform Bakhvalov-type mesh. For time discretization, the Crank-Nicolson scheme is applied separately in each spatial direction alongside the weak Galerkin method. Theoretical error analysis, based on a local (Formula presented.) projection, ensures spatial accuracy of order (Formula presented.) and second-order temporal convergence under the balanced norm. Numerical experiments validate the efficiency of the proposed method and confirm the theoretical error estimates.