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<p>In this paper, we introduce a Legendre spectral method that integrates domain decomposition and mapping techniques to address the Fokker-Planck equation on two-dimensional irregular domains. Initially, we partition the irregular domain into two subdomains and then transform these subdomains into regular ones through coordinate transformations. Subsequently, we utilize the classical Legendre spectral method for numerical simulations on these transformed regular subdomains. Our analysis focuses on the optimal rate of convergence of this method under the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2024_2776_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, with numerical results emphasizing the high accuracy of the proposed approach.</p>

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A Multi-domain Spectral Method with Mapping Techniques for the Fokker-Planck Equation in Irregular Domains

  • 期刊论文

  • 发表日期：

    2025年1月6日

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  • Chuan Wang,
  • Zicheng Wang,
  • Zhongqing Wang

摘要

In this paper, we introduce a Legendre spectral method that integrates domain decomposition and mapping techniques to address the Fokker-Planck equation on two-dimensional irregular domains. Initially, we partition the irregular domain into two subdomains and then transform these subdomains into regular ones through coordinate transformations. Subsequently, we utilize the classical Legendre spectral method for numerical simulations on these transformed regular subdomains. Our analysis focuses on the optimal rate of convergence of this method under the \(H^1\) H 1 -norm, with numerical results emphasizing the high accuracy of the proposed approach.