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<p>In this paper, a discontinuous Galerkin (DG) method is proposed to solve a first-order nonlinear singularly perturbed delay differential equation (SPDDE). At first, the construction of a generalized Bakhvalov-type (B-type) mesh and the corresponding characteristics are given. Furthermore, an optimal parameter-uniform convergence rate <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(s+1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>s</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> in the <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(L^2\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-norm is proved, where <i>s</i> is the degree of the piecewise polynomial space. Finally, two numerical experiments are carried out to complement the theoretical results of our proposed numerical method.</p>

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Convergence Analysis of a Discontinuous Galerkin Method on Bakhvalov-Type Meshes for a First-Order Nonlinear Singularly Perturbed Delay Problem

  • 期刊论文

  • 发表日期：

    2026年5月8日

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  • Yige Liao,
  • Li-Bin Liu,
  • Xianbing Luo

摘要

In this paper, a discontinuous Galerkin (DG) method is proposed to solve a first-order nonlinear singularly perturbed delay differential equation (SPDDE). At first, the construction of a generalized Bakhvalov-type (B-type) mesh and the corresponding characteristics are given. Furthermore, an optimal parameter-uniform convergence rate \(s+1\) s + 1 in the \(L^2\) L 2 -norm is proved, where s is the degree of the piecewise polynomial space. Finally, two numerical experiments are carried out to complement the theoretical results of our proposed numerical method.