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Ill‐conditioning in the virtual element method: Stabilizations and bases
Author(s) -
Mascotto Lorenzo
Publication year - 2018
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22257
Subject(s) - mathematics , monomial , element (criminal law) , condition number , stiffness matrix , order (exchange) , polynomial , matrix (chemical analysis) , poisson distribution , finite element method , mathematical analysis , pure mathematics , structural engineering , statistics , political science , law , economics , composite material , engineering , eigenvalues and eigenvectors , physics , materials science , finance , quantum mechanics
In this article, we investigate the behavior of the condition number of the stiffness matrix resulting from the approximation of a 2D Poisson problem by means of the virtual element method. It turns out that ill‐conditioning appears when considering high‐order methods or in presence of “bad‐shaped” (for instance nonuniformly star‐shaped, with small edges…) sequences of polygons. We show that in order to improve such condition number one can modify the definition of the internal moments by choosing proper polynomial functions that are not the standard monomials. We also give numerical evidence that at least for a 2D problem, standard choices for the stabilization give similar results in terms of condition number.