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<p>In this paper, we present the implementation details of the three-dimensional Nonconforming Virtual Element Method, building upon the theoretical framework introduced in (<i>ESAIM: Math. Model. Numer. Anal.</i>, <i>50</i>(3), 879–904,&#xa0;(2016), (<i>IMA J. Numer. Anal.</i>, <i>37</i>, (2015). In this formulation, the local spaces are suitably enhanced to allow the construction of the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\varvec{L}^2\)</EquationSource> </InlineEquation> projection of virtual functions onto polynomials of degree&#xa0;<InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\varvec{k}\)</EquationSource> </InlineEquation>. This higher-order projection provides a more accurate representation of the load term and, more importantly, enables the definition of the reaction term. We perform a numerical convergence study of the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\varvec{L}^2\)</EquationSource> </InlineEquation>-norm and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\varvec{H}^1\)</EquationSource> </InlineEquation>-seminorm errors with respect to both mesh size and polynomial degree in the three-dimensional setting. Numerical results are then compared with those obtained using the conforming VEM, showing the consistency and computational performance of the proposed 3D implementation.</p>

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The 3D Nonconforming Virtual Element Method with enhanced spaces: Numerical investigation and implementation guide

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  • 出版模式：

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  • 发表日期：

    2025年12月5日

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  • Franco Dassi,
  • Manuel Trezzi

摘要

In this paper, we present the implementation details of the three-dimensional Nonconforming Virtual Element Method, building upon the theoretical framework introduced in (ESAIM: Math. Model. Numer. Anal., 50(3), 879–904, (2016), (IMA J. Numer. Anal., 37, (2015). In this formulation, the local spaces are suitably enhanced to allow the construction of the \(\varvec{L}^2\) projection of virtual functions onto polynomials of degree  \(\varvec{k}\) . This higher-order projection provides a more accurate representation of the load term and, more importantly, enables the definition of the reaction term. We perform a numerical convergence study of the \(\varvec{L}^2\) -norm and \(\varvec{H}^1\) -seminorm errors with respect to both mesh size and polynomial degree in the three-dimensional setting. Numerical results are then compared with those obtained using the conforming VEM, showing the consistency and computational performance of the proposed 3D implementation.