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<p>We study the linearized Crank-Nicolson (CN) finite element scheme of the Poisson-Nernst-Planck (PNP) systems. First of all, we introduce the Ritz projection and discrete Laplace operator, along with several refined estimations of the nonlinear terms. Subsequently, utilizing an inductive mathematical approach, we establish the unconditional convergence in terms of the <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2504_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">\(\boldsymbol{{l^\infty }({L^2})}\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2504_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="63" /> </InlineMediaObject> <EquationSource Format="TEX">\(\boldsymbol{{l^\infty }({H^1})}\)</EquationSource> </InlineEquation> norms, without imposing any constraints on the relationship between the time step and the mesh size. Then, we employ post-processing interpolation techniques to achieve superconvergence results, which improves the convergence rate of the <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12190_2025_2504_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="63" /> </InlineMediaObject> <EquationSource Format="TEX">\(\boldsymbol{{l^\infty }({H^1})}\)</EquationSource> </InlineEquation> -norm by one order. Finally, we validate the theoretical analysis through a numerical example. This paper presents a novel approach to derive unconditional error estimations for the CN scheme of the PNP systems, which is more concise than previous error splitting techniques.</p>

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Superconvergence analysis for the Crank-Nicolson scheme of the Poisson-Nernst-Planck systems

  • 期刊论文

  • 发表日期：

    2025年5月6日

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  • Minghao Li,
  • Zijue Wang,
  • Liuchao Xiao

摘要

We study the linearized Crank-Nicolson (CN) finite element scheme of the Poisson-Nernst-Planck (PNP) systems. First of all, we introduce the Ritz projection and discrete Laplace operator, along with several refined estimations of the nonlinear terms. Subsequently, utilizing an inductive mathematical approach, we establish the unconditional convergence in terms of the \(\boldsymbol{{l^\infty }({L^2})}\) and \(\boldsymbol{{l^\infty }({H^1})}\) norms, without imposing any constraints on the relationship between the time step and the mesh size. Then, we employ post-processing interpolation techniques to achieve superconvergence results, which improves the convergence rate of the \(\boldsymbol{{l^\infty }({H^1})}\) -norm by one order. Finally, we validate the theoretical analysis through a numerical example. This paper presents a novel approach to derive unconditional error estimations for the CN scheme of the PNP systems, which is more concise than previous error splitting techniques.