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<p>In this paper, we present an optimal convergence analysis for virtual element method (VEM) of strongly nonlinear parabolic problems on general polygonal meshes. A linearized second-order backward difference formula (BDF2) is applied for the time discretization. To address the challenges associated with the nonlinear capacity term and the diffusion coefficient, we introduce a temporal-spatial error splitting technique that separates errors in the temporal and spatial directions and employs two virtual element projection operators. With the help of the time-discrete system, we establish the unconditional boundedness of the numerical solution in the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_3054_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="26" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^{\infty }\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>∞</mi> </msup> </math></EquationSource> </InlineEquation>-norm and prove optimal error estimates without any time-step restrictions. Numerical experiments are provided to confirm the theoretical results.</p>

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Convergence Analysis of Linearized BDF2 Virtual Element Method for Strongly Nonlinear Parabolic Problems on General Polygonal Meshes

  • 期刊论文

  • 发表日期：

    2025年9月12日

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  • Yang Wang,
  • Huaming Yi,
  • Xiaohui Wu,
  • Wanxiang Liu

摘要

In this paper, we present an optimal convergence analysis for virtual element method (VEM) of strongly nonlinear parabolic problems on general polygonal meshes. A linearized second-order backward difference formula (BDF2) is applied for the time discretization. To address the challenges associated with the nonlinear capacity term and the diffusion coefficient, we introduce a temporal-spatial error splitting technique that separates errors in the temporal and spatial directions and employs two virtual element projection operators. With the help of the time-discrete system, we establish the unconditional boundedness of the numerical solution in the \(L^{\infty }\) L ∞ -norm and prove optimal error estimates without any time-step restrictions. Numerical experiments are provided to confirm the theoretical results.