A Crank-Nicolson WG-FEM for unsteady 2D convection-diffusion equation with nonlinear reaction term on layer adapted mesh

Kumar N., TOPRAKSEVEN Ş., Singh Yadav N., Yuan J.

Applied Numerical Mathematics, vol.201, pp.322-346, 2024 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 201
  • Publication Date: 2024
  • Doi Number: 10.1016/j.apnum.2024.03.013
  • Journal Name: Applied Numerical Mathematics
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Applied Science & Technology Source, Compendex, Computer & Applied Sciences, INSPEC, MathSciNet, zbMATH, DIALNET
  • Page Numbers: pp.322-346
  • Keywords: Crank-Nicolson scheme, Semidiscrete, and fully discrete schemes, Shishkin mesh, Singularly perturbed convection-diffusion equation with a nonlinear reaction term, Weak Galerkin method, ε-uniform convergence
  • Yozgat Bozok University Affiliated: Yes

Abstract

This paper is contributed to explore how a Crank-Nicolson weak Galerkin finite element method (WG-FEM) addresses the singularly perturbed unsteady convection-diffusion equation with a nonlinear reaction term in 2D. The problem and some asymptotic behavior results are given for the exact solution and its derivatives with the parameter ε. These results are essential for proving the uniform convergence of the proposed WG-FEM. A tensor product Shishkin mesh featuring piece-wise discontinuous bilinear polynomials is employed to handle the layers for uniform convergence at different parameter values of ε. An error estimate ‖uN−INu‖WG is presented, where INu denotes the vertex-edge-cell interpolation of the solution u, and ‖⋅‖WG denotes the Weak Galerkin norm. The optimal uniform convergence order is demonstrated for semi-discrete and fully discrete schemes. Various numerical experiments are conducted to validate the optimal order of convergence demonstrated by the proposed method.