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Conforming Virtual Element Methods for Sobolev Equations

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Abstract

The intention of this paper is to study the \(H^1\)-conforming virtual element method for a class of Sobolev equations with variable coefficients. The semi-discrete scheme is constructed on the basis of the symmetric Nitsche’s method to impose the homogeneous and inhomogeneous Dirichlet boundary conditions in a unified way. The fully discrete scheme is obtained with the Crank–Nicolson scheme for temporal approximation. Under some assumptions about the penalty parameter in the Nitsche’s method, the existence and uniqueness of the semi-discrete and fully discrete solutions are analyzed. Furthermore, for the semi-discrete and fully discrete schemes, optimal a priori error estimates in both a mesh size dependent norm and \(L^2\)-norm are proven. Finally, some numerical experiments are carried out to illustrate the theoretical results.

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Acknowledgements

The authors are grateful to the referees for their valuable comments.

Funding

This work is supported by the National Natural Science Foundation of China (11771112, 12071100).

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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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Correspondence to Jingjun Zhao.

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Xu, Y., Zhou, Z. & Zhao, J. Conforming Virtual Element Methods for Sobolev Equations. J Sci Comput 93, 32 (2022). https://doi.org/10.1007/s10915-022-01997-3

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  • DOI: https://doi.org/10.1007/s10915-022-01997-3

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