On Arbitrarily Regular Conforming Virtual Element Methods for Elliptic Partial Differential Equations
Contributo in Atti di convegno
Data di Pubblicazione:
2023
Citazione:
On Arbitrarily Regular Conforming Virtual Element Methods for Elliptic Partial Differential Equations / P.F. Antonietti, G. Manzini, S. Scacchi, M. Verani (LECTURE NOTES IN COMPUTATIONAL SCIENCE AND ENGINEERING). - In: Spectral and High Order Methods for Partial Differential Equations ICOSAHOM / [a cura di] Melenk, J.M., Perugia, I., Schöberl, J., Schwab, C.. - Cham : Springer, 2023. - ISBN 978-3-031-20431-9. - pp. 3-30 (( Intervento presentato al 13. convegno International Conference on Spectral and High Order Methods, ICOSAHOM tenutosi a Vienna : 12-16 luglio nel 2021 [10.1007/978-3-031-20432-6_1].
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 Hp2(Ω), where p1 and p2 are two integer numbers such that p2 ≥ p1 ≥ 1 and Ω ⊂ ℝ2 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.
Tipologia IRIS:
03 - Contributo in volume
Elenco autori:
P.F. Antonietti, G. Manzini, S. Scacchi, M. Verani
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Titolo del libro:
Spectral and High Order Methods for Partial Differential Equations ICOSAHOM