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<p>In this paper, a weak Galerkin spectral element method is developed to solve non-layered convection–diffusion–reaction equations with variable coefficients, and a rigorous <i>hp</i>-error analysis is conducted. First, we construct the approximation spaces of weak gradients and weak functions by using a one-to-one mapping from the reference element to a physical element. Next, we establish a nonsymmetric weak Galerkin spectral element approximation scheme for convection–diffusion–reaction problems with variable coefficients, and prove its well-posedness. Furthermore, by exploiting the commutativity of projection operators and relevant approximation techniques, we derive the optimal error estimates on affine families of triangular and convex quadrilateral meshes with respect to the mesh size and the polynomial degree. Finally, numerical experiments are conducted on both triangular meshes and convex quadrilateral meshes to confirm the effectiveness of the proposed method for variable-coefficient convection–diffusion–reaction equations.</p>

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The weak Galerkin spectral element method for non-layered convection–diffusion–reaction equations with variable coefficients

  • 期刊论文

  • 发表日期：

    2026年7月10日

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  • Jiajia Pan,
  • Xiaofeng Xu

摘要

In this paper, a weak Galerkin spectral element method is developed to solve non-layered convection–diffusion–reaction equations with variable coefficients, and a rigorous hp-error analysis is conducted. First, we construct the approximation spaces of weak gradients and weak functions by using a one-to-one mapping from the reference element to a physical element. Next, we establish a nonsymmetric weak Galerkin spectral element approximation scheme for convection–diffusion–reaction problems with variable coefficients, and prove its well-posedness. Furthermore, by exploiting the commutativity of projection operators and relevant approximation techniques, we derive the optimal error estimates on affine families of triangular and convex quadrilateral meshes with respect to the mesh size and the polynomial degree. Finally, numerical experiments are conducted on both triangular meshes and convex quadrilateral meshes to confirm the effectiveness of the proposed method for variable-coefficient convection–diffusion–reaction equations.