Conforming and nonconforming virtual element methods for elliptic problems

By Andrea Cangiani, Gianmarco Manzini, Oliver J. Sutton. 2015

278 citations recorded by the source corpus.

We present in a unified framework new conforming and nonconforming Virtual Element Methods (VEM) for general second order elliptic problems in two and three dimensions. The differential operator is split into its symmetric and non-symmetric parts and conditions for stability and accuracy on their discrete counterparts are established. These conditions are shown to lead to optimal $H^1$- and $L^2$-error estimates, confirmed by numerical experiments on a set of polygonal meshes. The accuracy of the numerical approximation provided by the two methods is shown to be comparable.

Subfields: Computational Mechanics, Electrical and Electronic Engineering, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/fe93b1acb8d19ea8a7647dc005e6ab76e7761896

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

1. Giuseppe Vacca
2. David Mora
3. Iv'an Vel'asquez
4. Francesca Gardini
5. Gianmarco Manzini
6. L. Beirão da Veiga
7. Paola Francesca Antonietti
8. Alessandro Russo

In Electrical and Electronic Engineering

1. David Mora
2. Iv'an Vel'asquez
3. Giuseppe Vacca
4. Francesca Gardini
5. Gianmarco Manzini

In Mechanics of Materials

1. Giuseppe Vacca
2. David Mora
3. Iv'an Vel'asquez
4. Francesca Gardini
5. Gianmarco Manzini
6. L. Beirão da Veiga
7. Alessandro Russo

Citations in the corpus

• Virtual element for the buckling problem of Kirchhoff–Love plates (2019, background): “in a large range of problems in fluid and solid mechanics; see for instance [2, 3, 4, 7, 8, 9, 11, 12, 15, 16, 21, 22, 23, 24, 33, 40, 43, 44].”
• A residual a posteriori error estimate for the Virtual Element Method (2017, unknown): “VEM discretization The Virtual Element Method is a quite recent discretization approach and its extensions to more complex models than the one considered here are currently largely investigated.(35,5,12) Let Th be a discretization of Ω ⊂ R(2) with open star-shaped polygons having an arbitrary number”
• The nonconforming Virtual Element Method for eigenvalue problems (2018, methodology): “ to a certain order, of the jumps of the discrete space functions across all mesh interfaces are zero. The nonconforming VEM has been applied successfully to the general second-order elliptic problem [27], the Stokes equation [25], the biharmonic problem [5], and the nonconforming approach has been ”
• Orthogonal polynomials in badly shaped polygonal elements for the Virtual Element Method (2017, unknown): “Following [38, 39], we introduce the local finite dimensional space V E h := { v ∈ H(1) (E) : ∆v ∈ Pk (E) , v ∈ Pk (e) ∀e ⊂ ∂E, (v, p)E = ( Πk v, p ) E ∀p ∈ Pk (E) /Pk−2 (E) } , where the space Pk (E) /Pk−2 (E) is intended to contain those polynomials belonging to Pk (E) that are L(2) (E)-orthogonal”
• Virtual element method for quasilinear elliptic problems (2017, unknown): “retisation of problem (1.3), using general polygonal and polyhedral decompositions of in two and three dimensions, respectively. We start by recalling the denition of the virtual element spaces from [1, 17]. 2.1. The Discrete Spaces. The denition of the virtual element method relies on the availabil”