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<p>We explore the potential applications of virtual elements for solving the Sobolev equation with a convective term. A conforming virtual element method is employed for spatial discretization, while an implicit Euler scheme is used to approximate the time derivative. To establish the optimal rate of convergence, a novel intermediate projection operator is introduced. We discuss and analyze both the semi-discrete and fully discrete schemes, deriving optimal error estimates for both the energy norm and <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2587_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^2\)</EquationSource> </InlineEquation>-norm. Several numerical experiments are conducted to validate the theoretical findings and assess the computational efficiency of the proposed numerical methods.</p>

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Convergence analysis of virtual element methods for the Sobolev equation with convection

  • 期刊论文

  • 发表日期：

    2025年7月18日

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  • Ankit Kumar,
  • Sarvesh Kumar,
  • Sangita Yadav

摘要

We explore the potential applications of virtual elements for solving the Sobolev equation with a convective term. A conforming virtual element method is employed for spatial discretization, while an implicit Euler scheme is used to approximate the time derivative. To establish the optimal rate of convergence, a novel intermediate projection operator is introduced. We discuss and analyze both the semi-discrete and fully discrete schemes, deriving optimal error estimates for both the energy norm and \(L^2\) -norm. Several numerical experiments are conducted to validate the theoretical findings and assess the computational efficiency of the proposed numerical methods.