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Abstract <p>The null field method (NFM) has been applied successfully to Laplace’s equation in circular/elliptic domains with multiple circular/elliptic holes. There are many papers published; but no strict error analysis exists so far. In this paper, we describe the NFM as the Galerkin methods involving the trapezoidal rule. For the NFM, the pseudo-boundaries can be just located on the domain boundary <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11470_2025_2170_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Gamma \)</EquationSource> <!--ComMat2470190Lee-m1--> </InlineEquation>; this is the most intriguing and important characteristic in applications. In such a case, the error bounds are derived for the Dirichlet problems. Polynomial convergence rates are obtained, and exponential convergence rates can be achieved for infinite smooth solutions. Although the error analysis in this paper is made for the source nodes on <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11470_2025_2170_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Gamma \)</EquationSource> <!--ComMat2470190Lee-m2--> </InlineEquation>, it can be extended for the source nodes outside <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11470_2025_2170_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">\(\Gamma \)</EquationSource> <!--ComMat2470190Lee-m3--> </InlineEquation>. The error analysis in this paper is essential to the NFM because it provides some important theoretical foundation, thus to enhance its application.</p>

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Error Analysis of the Null Field Method for Laplace’s Equation on Circular Domains with Circular Holes: Source Nodes Just on Domain Boundaries

  • 期刊论文

  • 发表日期：

    2025年1月1日

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  • Ming-Gong Lee,
  • Hung-Tsai Huang,
  • Zi-Cai Li,
  • A. L. Kazakov

摘要

Abstract

The null field method (NFM) has been applied successfully to Laplace’s equation in circular/elliptic domains with multiple circular/elliptic holes. There are many papers published; but no strict error analysis exists so far. In this paper, we describe the NFM as the Galerkin methods involving the trapezoidal rule. For the NFM, the pseudo-boundaries can be just located on the domain boundary \(\Gamma \) ; this is the most intriguing and important characteristic in applications. In such a case, the error bounds are derived for the Dirichlet problems. Polynomial convergence rates are obtained, and exponential convergence rates can be achieved for infinite smooth solutions. Although the error analysis in this paper is made for the source nodes on \(\Gamma \) , it can be extended for the source nodes outside \(\Gamma \) . The error analysis in this paper is essential to the NFM because it provides some important theoretical foundation, thus to enhance its application.