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Error Estimates of Conforming Virtual Element Methods with a Modified Symmetric Nitsche’s Formula for 2D Semilinear Parabolic Equations

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Abstract

In this paper, we study the \(H^1\)-conforming virtual element method for the spatial discretization of semilinear parabolic equations with inhomogeneous Dirichlet boundary conditions based on a modified symmetric Nitsche’s formula. Herein, the modified Nitsche’s formula is constructed by introducing a global lifting operator which maps the trace of an \(H^1\)-function into the global conforming virtual element space. In contrast to the classical symmetric Nitsche’s method, the penalty parameter in the modified symmetric Nitsche’s method does not need to be greater than a strictly positive lower bound, and it only needs to be greater than 0 to provide a coercive spatial bilinear form. For time discretization, the second-order backward difference formula is used. On the basis of some assumptions on the given problem data, the optimal error estimates in both an energy norm and \(L^2\)-norm are established for the semi-discrete and fully discrete schemes. The theoretical results are verified by some numerical experiments.

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Acknowledgements

The authors are greatly indebted to the referees for useful comments.

Funding

This work is supported by the National Natural Science Foundation of China (Project Numbers: 11771112, 12071100) and the Fundamental Research Funds for the Central Universities (Project Number: 2022FRFK060019).

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Authors and Affiliations

  1. School of Mathematics, Harbin Institute of Technology, Harbin, 150001, China

    Yang Xu, Zhenguo Zhou & Jingjun Zhao

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  1. Yang Xu
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  2. Zhenguo Zhou
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  3. Jingjun Zhao
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Correspondence to Jingjun Zhao.

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Xu, Y., Zhou, Z. & Zhao, J. Error Estimates of Conforming Virtual Element Methods with a Modified Symmetric Nitsche’s Formula for 2D Semilinear Parabolic Equations. J Sci Comput 95, 69 (2023). https://doi.org/10.1007/s10915-023-02188-4

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  • DOI: https://doi.org/10.1007/s10915-023-02188-4

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