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<p>In this paper, we develop an efficient Fourier-Legendre spectral-Galerkin method for solving elliptic partial differential equations on general two-dimensional domains. A key core of our approach is employing a harmonic map to handle the general physical domains. This technique ensures broad geometric applicability, making the method highly effective for both complex star-shaped and nonstar-shaped domains. Moreover, this method is rigorously proved, with optimal convergence results established under <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(H^1\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>H</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-norm, which is independent of the domain boundary’s smoothness. The effectiveness and generality of the scheme are validated through some numerical examples on a wide variety of complex geometries.</p>

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A Spectral Method with Harmonic Map for Elliptic PDEs on General Two-Dimensional Domains

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年3月31日

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  • Shan Shi,
  • Xiaoyun Jiang,
  • Fanhai Zeng,
  • Hui Zhang

摘要

In this paper, we develop an efficient Fourier-Legendre spectral-Galerkin method for solving elliptic partial differential equations on general two-dimensional domains. A key core of our approach is employing a harmonic map to handle the general physical domains. This technique ensures broad geometric applicability, making the method highly effective for both complex star-shaped and nonstar-shaped domains. Moreover, this method is rigorously proved, with optimal convergence results established under \(H^1\) H 1 -norm, which is independent of the domain boundary’s smoothness. The effectiveness and generality of the scheme are validated through some numerical examples on a wide variety of complex geometries.