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Journal of Scientific Computing

, Volume 77, Issue 3, pp 1874–1908 | Cite as

Non-conforming Harmonic Virtual Element Method: \(h\)- and \(p\)-Versions

  • Lorenzo Mascotto
  • Ilaria Perugia
  • Alexander Pichler
Article
  • 50 Downloads

Abstract

We study the \(h\)- and \(p\)-versions of non-conforming harmonic virtual element methods (VEM) for the approximation of the Dirichlet–Laplace problem on a 2D polygonal domain, providing quasi-optimal error bounds. Harmonic VEM do not make use of internal degrees of freedom. This leads to a faster convergence, in terms of the number of degrees of freedom, as compared to standard VEM. Importantly, the technical tools used in our \(p\)-analysis can be employed as well in the analysis of more general non-conforming finite element methods and VEM. The theoretical results are validated in a series of numerical experiments. The hp-version of the method is numerically tested, demonstrating exponential convergence with rate given by the square root of the number of degrees of freedom.

Keywords

Virtual element methods Non-conforming methods Laplace problem Approximation by harmonic functions hp error bounds Polytopal meshes 

Mathematics Subject Classification

65N30 65N12 65N15 35J05 31A05 

Notes

Acknowledgements

The authors have been funded by the Austrian Science Fund (FWF) through the projects P 29197-N32 and F 65. They are very grateful to the anonymous referees for their valuable and constructive comments, which have contributed to the improvement of the paper.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculty of MathematicsUniversity of ViennaViennaAustria

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