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Abstract <p>A numerical solution by the finite element method of a homogeneous Dirichlet boundary value problem for an elliptic equation is examined (using a Poisson equation as an example) in a two-dimensional convex polygonal domain <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\Omega \)</EquationSource> <!--NumAnAp2570004Romanov-m1--> </InlineEquation> with a singular right-hand side given by the Dirac delta function. A theorem on the existence and uniqueness of a generalized solution in the fractional Sobolev space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\({{H}^{s}}(\Omega )\)</EquationSource> <!--NumAnAp2570004Romanov-m2--> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(1{\text{/}}2 &lt; s &lt; 1\)</EquationSource> <!--NumAnAp2570004Romanov-m3--> </InlineEquation>, is proved. An approach to discrete analysis of the problem using the finite element method is proposed and investigated. The results of numerical experiments for a model problem, obtained using the FreeFem++ software, are presented. They confirm the error estimate of the difference between the discrete and exact solutions derived in the paper.</p>

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Finite Element Method Solution of a Boundary Value Problem for an Elliptic Equation with the Dirac Delta Function on the Right-Hand Side

  • 期刊论文

  • 发表日期：

    2025年12月1日

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  • D. N. Romanov,
  • M. V. Urev

摘要

Abstract

A numerical solution by the finite element method of a homogeneous Dirichlet boundary value problem for an elliptic equation is examined (using a Poisson equation as an example) in a two-dimensional convex polygonal domain \(\Omega \) with a singular right-hand side given by the Dirac delta function. A theorem on the existence and uniqueness of a generalized solution in the fractional Sobolev space \({{H}^{s}}(\Omega )\) , \(1{\text{/}}2 < s < 1\) , is proved. An approach to discrete analysis of the problem using the finite element method is proposed and investigated. The results of numerical experiments for a model problem, obtained using the FreeFem++ software, are presented. They confirm the error estimate of the difference between the discrete and exact solutions derived in the paper.