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

Xiaoqi Ma, Jin Zhang, Xinyi Feng, Chunxiao Zhang

TL;DR

The paper addresses the numerical approximation of a singularly perturbed 1D convection–diffusion problem using a direct discontinuous Galerkin (DDG) method on a Shishkin mesh. It introduces a composite interpolation that uses a global projection outside the layer and a Gauß–Lobatto projection inside the layer, and it selects numerical flux parameters to achieve almost $k+1$ order supercloseness in the energy norm $\|\cdot\|_E$. The analysis develops regularity results, mesh construction, DDG formulation, interpolation error bounds, and a detailed supercloseness proof, validated by numerical experiments. The approach provides stable, layer-resolving convergence with moderate mesh refinement, offering a practical tool for singular perturbation convection–diffusion problems.

Abstract

This paper investigates the supercloseness of a singularly perturbed convection diffusion problem using the direct discontinuous Galerkin (DDG) method on a Shishkin mesh. The main technical difficulties lie in controlling the diffusion term inside the layer, the convection term outside the layer, and the inter element jump term caused by the discontinuity of the numerical solution. The main idea is to design a new composite interpolation, in which a global projection is used outside the layer to satisfy the interface conditions determined by the selection of numerical flux, thereby eliminating or controlling the troublesome terms on the unit interface; and inside the layer, Gauß Lobatto projection is used to improve the convergence order of the diffusion term. On the basis of that, by selecting appropriate parameters in the numerical flux, we obtain the supercloseness result of almost $k+1$ order under an energy norm. Numerical experiments support our main theoretical conclusion.

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

TL;DR

The paper addresses the numerical approximation of a singularly perturbed 1D convection–diffusion problem using a direct discontinuous Galerkin (DDG) method on a Shishkin mesh. It introduces a composite interpolation that uses a global projection outside the layer and a Gauß–Lobatto projection inside the layer, and it selects numerical flux parameters to achieve almost order supercloseness in the energy norm . The analysis develops regularity results, mesh construction, DDG formulation, interpolation error bounds, and a detailed supercloseness proof, validated by numerical experiments. The approach provides stable, layer-resolving convergence with moderate mesh refinement, offering a practical tool for singular perturbation convection–diffusion problems.

Abstract

This paper investigates the supercloseness of a singularly perturbed convection diffusion problem using the direct discontinuous Galerkin (DDG) method on a Shishkin mesh. The main technical difficulties lie in controlling the diffusion term inside the layer, the convection term outside the layer, and the inter element jump term caused by the discontinuity of the numerical solution. The main idea is to design a new composite interpolation, in which a global projection is used outside the layer to satisfy the interface conditions determined by the selection of numerical flux, thereby eliminating or controlling the troublesome terms on the unit interface; and inside the layer, Gauß Lobatto projection is used to improve the convergence order of the diffusion term. On the basis of that, by selecting appropriate parameters in the numerical flux, we obtain the supercloseness result of almost order under an energy norm. Numerical experiments support our main theoretical conclusion.
Paper Structure (17 sections, 11 theorems, 98 equations, 1 table)

This paper contains 17 sections, 11 theorems, 98 equations, 1 table.

Table of Contents

  1. Introduction
  2. Regularity of the solution and Shishkin mesh
  3. A-$prior$ information of the solution
  4. Shishkin mesh
  5. The DDG method
  6. The DDG method
  7. The admissibility of DDG method
  8. Galerkin orthogonality and stability of the DDG method
  9. Interpolation and convergence analysis
  10. Interpolation
  11. Interpolation error
  12. Supercloseness
  13. Numerical experiments
  14. Statements and Declarations
  15. Funding
  16. ...and 2 more sections

Key Result

Theorem 1

Suppose that the condition eq:SPP-condition-1 holds and the functions $a, b, f$ are sufficiently smooth. Then the solution $w$ of eq:SPP-1d admits the decomposition where $S$ and $E$ meet $LS=f$ and $LE=0$, respectively. Moreover, for $0\le i\le k+1$, there is Note that here $k$ depends on the regularity of the coefficients, and when $a, b, f\in C^{\infty}(\Theta)$, eq:regularity holds.

Theorems & Definitions (26)

  • Theorem 1
  • proof
  • Definition 1
  • Lemma 1
  • proof
  • Corollary 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 16 more