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Supercloseness of the DDG method for a singularly perturbed convection diffusion problem on Shishkin mesh

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  • Ma, Xiaoqi
  • Zhang, Jin
  • Feng, Xinyi
  • Zhang, Chunxiao

Abstract

This paper investigates the supercloseness of a singularly perturbed convection diffusion problem discretized by the direct discontinuous Galerkin (DDG) method on a Shishkin mesh. The main technical challenges involve controlling the diffusion term inside the layer, the convection term outside the layer, and the inter-element jump terms induced by the discontinuity of the numerical solution. To address these issues, we design a new composite interpolation. Outside the layer, a global projection is employed to satisfy the interface conditions imposed by the numerical flux, which helps eliminate or control the problematic terms across element interfaces. Inside the layer, the Gauß-Lobatto projection is adopted to enhance the convergence order of the diffusion term. Based on this interpolation and an appropriate choice of parameters in the numerical flux, we derive a supercloseness result of order nearly k+12 when the perturbation parameter ϵ≤N−1, which improves to order k+1 in an energy norm for ϵ≤N−2. Numerical experiments provided to support the theoretical findings.

Suggested Citation

  • Ma, Xiaoqi & Zhang, Jin & Feng, Xinyi & Zhang, Chunxiao, 2026. "Supercloseness of the DDG method for a singularly perturbed convection diffusion problem on Shishkin mesh," Applied Mathematics and Computation, Elsevier, vol. 516(C).
    • Handle: RePEc:eee:apmaco:v:516:y:2026:i:c:s0096300325006071
    DOI: 10.1016/j.amc.2025.129882
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