Stability Analysis for the Virtual Element Method

By L. Beirão da Veiga, Carlo Lovadina, Alessandro Russo. 2016

311 citations recorded by the source corpus.

We analyse the Virtual Element Methods (VEM) on a simple elliptic model problem, allowing for more general meshes than the one typically considered in the VEM literature. For instance, meshes with arbitrarily small edges (with respect to the parent element diameter), can be dealt with. Our general approach applies to different choices of the stability form, including, for example, the "classical" one introduced in [L. Beirao da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini, and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013), no. 1, 199-214], and a recent one presented in [Wriggers, P., Rust, W.T., and Reddy, B.D., A virtual element method for contact, submitted for publication]. Finally, we show that the stabilization term can be simplified by dropping the contribution of the internal-to-the-element degrees of freedom. The resulting stabilization form, involving only the boundary degrees of freedom, can be used in the VEM scheme without affecting the stability and convergence properties. The numerical tests are in accordance with the theoretical predictions.

Subfields: Computational Mechanics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/4d3aab2eb8551968bc51aa0c6369768d032668c6

Citations in the corpus

• Orthogonal polynomials in badly shaped polygonal elements for the Virtual Element Method (2017, unknown): “…that the proposed method aims at improving the reliability of the computations performed in the set up of the consistent part of the VEM formulation of the problem and is completely independent of the VEM stabilization that is added to the consistent part in order to get a well posed problem [36].”