Virtual Element Methods for general second order elliptic problems on polygonal meshes

By L. BEIR˜AO, Davi Veiga, Franco Brezzi, L. Donatella Marini, Alessandro Russo. 2014

466 citations recorded by the source corpus.

We consider the discretization of a boundary value problem for a general linear second-order elliptic operator with smooth coefficients using the Virtual Element approach. As in [59] the problem is supposed to have a unique solution, but the associated bilinear form is not supposed to be coercive. Contrary to what was previously done for Virtual Element Methods (as for instance in [9]), we use here, in a systematic way, the L^2-projection operators as designed in [1]. In particular, the present method does not reduce to the original Virtual Element Method of [9] for simpler problems as the classical Laplace operator (apart from the lowest order cases). Numerical experiments show the accuracy and the robustness of the method, and they show as well that a simple-minded extension of the method in [9] to the case of variable coefficients produces, in general, sub-optimal results.

Subfields: Computational Mechanics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/7dbe022ff97d191379f5a31f00c4b7b108ed9708

How researchers see this paper

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In Computational Mechanics

1. Lorenzo Mascotto
2. Paola Francesca Antonietti
3. Marco Verani

In Numerical Analysis

1. Lorenzo Mascotto
2. Paola Francesca Antonietti
3. Marco Verani

Citations in the corpus

• A multigrid algorithm for the $p$-version of the Virtual Element Method (2017, methodology): “Recently, in [17] the virtual element method (VEM) has been introduced, and further developed for elliptic and parabolic problems in [22, 24].”
• Serendipity Virtual Elements for General Elliptic Equations in Three Dimensions (2018, unknown): “We here mention, in addition to the ones above, a sample of papers [1, 4, 8, 13, 17, 19, 22, 27] and refer to [25] for a more complete survey of the existing VEM literature.”
• Orthogonal polynomials in badly shaped polygonal elements for the Virtual Element Method (2017, unknown): “Following [38, 39], we introduce the local finite dimensional space where the space We then define the global Virtual Element Space on T h by gluing local spaces asking for continuity: The following degrees of freedom are unisolvent for V h (see [37, 38]): 1. the values at the vertices of the polygon;”
• Conforming and nonconforming virtual element methods for elliptic problems (2015, unknown): “Indeed, for each mesh shown, the difference on the number of degrees of freedom does not depend on the polynomial degree k and is about equal to the number of elements, in favour of the conforming VEM.”