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<p>In this paper, we introduce a new spherical coordinate transformation, which transforms three-dimensional curved geometries into a unit sphere. This transformation plays an important role in spectral approximations of differential equations in three-dimensional curved geometries. Some basic properties of the spherical coordinate transformation are given. As examples, we consider an elliptic equation in three-dimensional curved geometries, prove the existence and uniqueness of the weak solution, construct the Fourier-Legendre spectral-Galerkin scheme and analyze the optimal convergence of numerical solutions under <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_2919_Article_IEq1.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. We also apply the suggested approach to the Gross-Pitaevskii equation in three-dimensional curved geometries and present some numerical results. The proposed algorithm is very effective and easy to implement for problems in three-dimensional curved geometries. Abundant numerical results show that our spectral-Galerkin method possesses high order accuracy.</p>

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Efficient Spectral Methods for PDEs in Three-Dimensional Curved Geometries

  • 期刊论文

  • 发表日期：

    2025年5月13日

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  • Guoqing Yao,
  • Xian Wen,
  • Zhongqing Wang

摘要

In this paper, we introduce a new spherical coordinate transformation, which transforms three-dimensional curved geometries into a unit sphere. This transformation plays an important role in spectral approximations of differential equations in three-dimensional curved geometries. Some basic properties of the spherical coordinate transformation are given. As examples, we consider an elliptic equation in three-dimensional curved geometries, prove the existence and uniqueness of the weak solution, construct the Fourier-Legendre spectral-Galerkin scheme and analyze the optimal convergence of numerical solutions under \(H^1\) H 1 -norm. We also apply the suggested approach to the Gross-Pitaevskii equation in three-dimensional curved geometries and present some numerical results. The proposed algorithm is very effective and easy to implement for problems in three-dimensional curved geometries. Abundant numerical results show that our spectral-Galerkin method possesses high order accuracy.