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<p>Linearization is a valid approach to improve the computational efficiency of numerical methods for nonlinear partial differential equations. A linearized nonconforming virtual element method is proposed for the semilinear Sobolev equations in this work, where the linearized Euler backward scheme is employed to discretize the temporal variable. Both the semi-discrete scheme and fully discrete scheme are established and analyzed. By employing a Ritz projection operator, the optimal convergence orders in broken H<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2024_2333_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="8" /> </InlineMediaObject> <EquationSource Format="TEX">\(^1\)</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow /> <mrow /> <mn>1</mn> </mmultiscripts> </math></EquationSource> </InlineEquation> semi-norm and L<InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2024_2333_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="8" /> </InlineMediaObject> <EquationSource Format="TEX">\(^2\)</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow /> <mrow /> <mn>2</mn> </mmultiscripts> </math></EquationSource> </InlineEquation> norm are both derived. In the end, three numerical examples are conducted to inspect the correctness of theoretical analysis results.</p>

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Linearized nonconforming virtual element method for the semilinear Sobolev equations

  • 期刊论文

  • 发表日期：

    2025年2月21日

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  • Buying Zhang,
  • Wenhao Zhu,
  • Jikun Zhao

摘要

Linearization is a valid approach to improve the computational efficiency of numerical methods for nonlinear partial differential equations. A linearized nonconforming virtual element method is proposed for the semilinear Sobolev equations in this work, where the linearized Euler backward scheme is employed to discretize the temporal variable. Both the semi-discrete scheme and fully discrete scheme are established and analyzed. By employing a Ritz projection operator, the optimal convergence orders in broken H \(^1\) 1 semi-norm and L \(^2\) 2 norm are both derived. In the end, three numerical examples are conducted to inspect the correctness of theoretical analysis results.