The $p$- and $hp$-versions of the virtual element method for elliptic eigenvalue problems

We discuss the p- and hp-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 by product, 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
hp-refinements. Importantly, the geometric flexibility of polygonal meshes is exploited
in the construction of the hp-spaces.