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<p>The problem of studying the behaviour of solutions of boundary value problems for partial differential equations in domains with geometric singularities is of great importance in mathematics and its applications. In this paper, we consider as model problem the Dirichlet boundary value problem for the heat equation on polygonal domains and derive, on the one hand, explicit computable formulas for the singular solutions near non-convex corners. On the other hand, the formulas are used to construct an efficient adaptation for the Galerkin finite element method on quasiuniform meshes for the spatial discretization. For the temporal discretization, we employ the classical Crank-Nicolson method. A priori error estimates in the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11075_2025_2243_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(L_2\)</EquationSource> </InlineEquation>- and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11075_2025_2243_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="23" /> </InlineMediaObject> <EquationSource Format="TEX">\(H^1\)</EquationSource> </InlineEquation>-norms for both the semi-discrete as well as for the full discrete problems show that in non-convex domains the new adaptation exhibits the same order of accuracy as it is known for problems with regular solutions. Several numerical experiments are included to illustrate the practical implementation of the new algorithm and to confirm the theoretically expected rates of convergence. The analysis and numerical method presented herein would also apply to more general second order initial boundary value problems for parabolic or hyperbolic equations in polygonal domains.</p>

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A predictor-corrector finite element method for parabolic problems on polygonal domains

  • 期刊论文

  • 发表日期：

    2025年10月21日

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  • Boniface Nkemzi,
  • Jake Leonard Nkeck

摘要

The problem of studying the behaviour of solutions of boundary value problems for partial differential equations in domains with geometric singularities is of great importance in mathematics and its applications. In this paper, we consider as model problem the Dirichlet boundary value problem for the heat equation on polygonal domains and derive, on the one hand, explicit computable formulas for the singular solutions near non-convex corners. On the other hand, the formulas are used to construct an efficient adaptation for the Galerkin finite element method on quasiuniform meshes for the spatial discretization. For the temporal discretization, we employ the classical Crank-Nicolson method. A priori error estimates in the \(L_2\) - and \(H^1\) -norms for both the semi-discrete as well as for the full discrete problems show that in non-convex domains the new adaptation exhibits the same order of accuracy as it is known for problems with regular solutions. Several numerical experiments are included to illustrate the practical implementation of the new algorithm and to confirm the theoretically expected rates of convergence. The analysis and numerical method presented herein would also apply to more general second order initial boundary value problems for parabolic or hyperbolic equations in polygonal domains.