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<p>In this work, we construct a new numerical approach to solve two-dimensional elliptic singularly perturbed convection-diffusion problem (SPCDP) having regular boundary layers. The proposed approach combines the use of an adequate Shishkin mesh, which is specifically designed to resolve the type of regular boundary layers which appear in the exact solution, with a modified bi-cubic <i>B</i>-spline (MBCBS) collocation technique, that ensures high-order accuracy of the numerical method. Higher-order splines are particularly advantageous from a practical perspective, as it delivers accurate approximations without increase significantly the computational cost of the algorithm. A detailed analysis of error is performed, and therefore, the proposed method is proven to be a nearly second-order parameter uniformly convergent in the maximum norm. Numerical experiments are also provided to validate in practice the theoretical results and to corroborate the effectiveness and robustness of the proposed scheme on challenging test problems.</p>

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A Shishkin mesh based modified bi-cubic B-spline approach for two-dimensional elliptic convection-diffusion problems with regular boundary layers

  • 期刊论文

  • 发表日期：

    2026年3月9日

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  • Hemlata Jangid,
  • Parvin Kumari,
  • Carmelo Clavero

摘要

In this work, we construct a new numerical approach to solve two-dimensional elliptic singularly perturbed convection-diffusion problem (SPCDP) having regular boundary layers. The proposed approach combines the use of an adequate Shishkin mesh, which is specifically designed to resolve the type of regular boundary layers which appear in the exact solution, with a modified bi-cubic B-spline (MBCBS) collocation technique, that ensures high-order accuracy of the numerical method. Higher-order splines are particularly advantageous from a practical perspective, as it delivers accurate approximations without increase significantly the computational cost of the algorithm. A detailed analysis of error is performed, and therefore, the proposed method is proven to be a nearly second-order parameter uniformly convergent in the maximum norm. Numerical experiments are also provided to validate in practice the theoretical results and to corroborate the effectiveness and robustness of the proposed scheme on challenging test problems.