The p- and hp-versions of the virtual element method for elliptic eigenvalue problems. (1st April 2020)

• Record Type: Journal Article
• Authors: Čertík, O.; Gardini, F.; Manzini, G.; Mascotto, L.; Vacca, G.
• Is Part Of: Computers & mathematics with applications. Volume 79:issue 7(2020)
• Page Start: 2035
• Page End: 2056
• Publication Date: 2020-04-01
• Subjects: Virtual element methods -- Polygonal meshes -- Eigenvalue problems -- p- and hp-Galerkin methods; Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2019.10.018
• Languages: English
• ISSNs: 0898-1221

Abstract

Abstract: We discuss the p - and h p -versions of the virtual element method for the approximation of eigenpairs of elliptic operators with a potential term on polygonal meshes. An application of this model is provided by the Schrödinger equation with a pseudo-potential term. As an interesting byproduct, we present for the first time in literature an explicit construction of the stabilization of the mass matrix. We present in detail the analysis of the p -version of the method, proving exponential convergence in the case of analytic eigenfunctions. The theoretical results are supplied with a wide set of experiments. We also show numerically that, in the case of eigenfunctions with finite Sobolev regularity, an exponential approximation of the eigenvalues in terms of the cubic root of the number of degrees of freedom can be obtained by employing h p -refinements. Importantly, the geometric flexibility of polygonal meshes is exploited in the construction of the h p -spaces.