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On Arbitrarily Regular Conforming Virtual Element Methods for Elliptic Partial Differential Equations

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Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2020+1

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 137))

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Abstract

The Virtual Element Method (VEM) is a very effective framework to design numerical approximations with high global regularity to the solutions of elliptic partial differential equations. In this paper, we review the construction of such approximations for an elliptic problem of order 2p1 using conforming, finite dimensional subspaces of \(H^{p_2}(\Omega )\), where p1 and p2 are two integer numbers such that p2 ≥ p1 ≥ 1 and \(\Omega \subset \mathbb {R}^2\) is the computational domain. An abstract convergence result is presented in a suitably defined energy norm. The space formulation and major aspects such as the choice and unisolvence of the degrees of freedom are discussed, also providing specific examples corresponding to various practical cases of high global regularity. Finally, the construction of the “enhanced” formulation of the virtual element spaces is also discussed in details with a proof that the dimension of the “regular” and “enhanced” spaces is the same and that the virtual element functions in both spaces can be described by the same choice of the degrees of freedom.

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Acknowledgements

PFA and MV acknowledge the financial support of PRIN research grant number 201744KLJL “Virtual Element Methods: Analysis and Applications” and PRIN research grant number 20204LN5N5 “Advanced polyhedral discretisations of heterogeneous PDEs for multiphysics problems” funded by MIUR. GM acknowledges the financial support of the LDRD program of Los Alamos National Laboratory under project number 20220129ER. Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear Security Administration of U.S. Department of Energy (Contract No. 89233218CNA000001). The Authors are affiliated to GNCS-INdAM (Italy).

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Authors and Affiliations

  1. MOX, Dipartimento di Matematica, Politecnico di Milano, Milan, Italy

    P. F. Antonietti & M. Verani

  2. Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA

    G. Manzini

  3. Dipartimento di Matematica, Università degli Studi di Milano, Milan, Italy

    S. Scacchi

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  1. P. F. Antonietti
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  2. G. Manzini
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  3. S. Scacchi
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  4. M. Verani
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Correspondence to M. Verani .

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Editors and Affiliations

  1. TU Wien, Wien, Austria

    Jens M. Melenk

  2. Universität Wien, Wien, Austria

    Ilaria Perugia

  3. TU Wien, Wien, Austria

    Joachim Schöberl

  4. ETH Zürich, Zürich, Switzerland

    Christoph Schwab

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Antonietti, P.F., Manzini, G., Scacchi, S., Verani, M. (2023). On Arbitrarily Regular Conforming Virtual Element Methods for Elliptic Partial Differential Equations. In: Melenk, J.M., Perugia, I., Schöberl, J., Schwab, C. (eds) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2020+1. Lecture Notes in Computational Science and Engineering, vol 137. Springer, Cham. https://doi.org/10.1007/978-3-031-20432-6_1

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