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<p>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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> projection as an intermediate operator for each spatial variable. Our main findings indicate that the proposed method achieves a convergence order of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal {O}(N^{-k})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>N</mi> <mrow> <mo>-</mo> <mi>k</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> 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.</p>

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

  • 期刊论文

  • 发表日期：

    2025年4月24日

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  • Naresh Kumar,
  • Jasbir Singh,
  • Şuayip Toprakseven,
  • Ram Jiwari

摘要

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 \(L^2\) L 2 projection as an intermediate operator for each spatial variable. Our main findings indicate that the proposed method achieves a convergence order of \(\mathcal {O}(N^{-k})\) 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.