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<p>This paper presents an efficient Fourier-Legendre-Jacobi rational spectral method, based on mapping techniques, for solving singularly perturbed convection-diffusion-reaction problems in a three-dimensional exterior domain with a complex obstacle. The solutions exhibit boundary or interior layer behavior as <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_3060_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">\(\epsilon \rightarrow 0.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ϵ</mi> <mo stretchy="false">→</mo> <mn>0</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> The method begins by applying a spherical coordinate transformation to map the exterior domain of the complex obstacle onto the exterior of a unit sphere, while simultaneously transforming the convection-diffusion-reaction equation. The transformed equation is then formulated in its weak form, and a Fourier-Legendre-Jacobi rational spectral scheme is introduced. The paper provides a detailed description of the numerical implementation and analyzes the convergence of the solution in the <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_3060_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">\(H^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-norm. Numerical results demonstrate that the proposed method achieves high-order accuracy.</p>

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An Efficient Spectral Method for Singularly Perturbed Convection-Diffusion-Reaction Problems in Three-Dimensional Irregular Exterior Domains

  • 期刊论文

  • 发表日期：

    2025年9月17日

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  • Chuan Wang,
  • Zicheng Wang,
  • Zhongqing Wang

摘要

This paper presents an efficient Fourier-Legendre-Jacobi rational spectral method, based on mapping techniques, for solving singularly perturbed convection-diffusion-reaction problems in a three-dimensional exterior domain with a complex obstacle. The solutions exhibit boundary or interior layer behavior as \(\epsilon \rightarrow 0.\) ϵ → 0 . The method begins by applying a spherical coordinate transformation to map the exterior domain of the complex obstacle onto the exterior of a unit sphere, while simultaneously transforming the convection-diffusion-reaction equation. The transformed equation is then formulated in its weak form, and a Fourier-Legendre-Jacobi rational spectral scheme is introduced. The paper provides a detailed description of the numerical implementation and analyzes the convergence of the solution in the \(H^1\) H 1 -norm. Numerical results demonstrate that the proposed method achieves high-order accuracy.