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<p>This paper investigates the weak Galerkin finite element method (WG-FEM) for a singularly perturbed fourth-order reaction-diffusion problem. Based on a solution decomposition and a Shishkin mesh, we establish a parameter-robust convergence of the WG-FEM. Specifically, we prove that the method achieves an error estimate of order <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\varvec{\mathcal {O}}(\varvec{\varepsilon }^{\varvec{1/2}}(\varvec{N}^{\varvec{-1}}\varvec{\ln N})^{\varvec{r-1}} \varvec{+ N}^{\varvec{-r}})\)</EquationSource> </InlineEquation> in the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\varvec{H^2}\)</EquationSource> </InlineEquation> discrete energy norm when piecewise polynomials of order <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\varvec{r \ge 2}\)</EquationSource> </InlineEquation> are used on the interior of the intervals and constant polynomials at the mesh points, where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\varvec{N}\)</EquationSource> </InlineEquation> is the number of subintervals. Numerical examples are presented to support the theoretical findings and confirm the sharpness of the error estimate.</p>

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Error estimates of the weak Galerkin method for singularly perturbed fourth-order reaction-diffusion problems

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月1日

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  • Suayip Toprakseven,
  • Srinivasan Natesan

摘要

This paper investigates the weak Galerkin finite element method (WG-FEM) for a singularly perturbed fourth-order reaction-diffusion problem. Based on a solution decomposition and a Shishkin mesh, we establish a parameter-robust convergence of the WG-FEM. Specifically, we prove that the method achieves an error estimate of order \(\varvec{\mathcal {O}}(\varvec{\varepsilon }^{\varvec{1/2}}(\varvec{N}^{\varvec{-1}}\varvec{\ln N})^{\varvec{r-1}} \varvec{+ N}^{\varvec{-r}})\) in the \(\varvec{H^2}\) discrete energy norm when piecewise polynomials of order \(\varvec{r \ge 2}\) are used on the interior of the intervals and constant polynomials at the mesh points, where \(\varvec{N}\) is the number of subintervals. Numerical examples are presented to support the theoretical findings and confirm the sharpness of the error estimate.