On arbitrarily regular conforming virtual element methods for elliptic partial differential equations

Gianmarco Manzini, Paola Francesca Antonietti, Simone Scacchi, Marco Verani

Publication date: 25 December 2021

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

p_{1}

using conforming, finite dimensional subspaces of

H^{p_{2}} (O m e g a)

, where

p_{1}

and

p_{2}

are two integer numbers such that

p_{2} g e q p_{1} g e q 1

and

O m e g a i n 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.

Mathematics Subject Classification ID

Error bounds for boundary value problems involving PDEs (65N15) Stability and convergence of numerical methods for boundary value problems involving PDEs (65N12)

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