The $p$- and $hp$-versions of the virtual element method for elliptic eigenvalue problems
This work provides a theoretical and numerical foundation for high-order virtual element methods in eigenvalue problems, enabling efficient approximation on polygonal meshes.
The paper extends the p- and hp-versions of the virtual element method to elliptic eigenvalue problems, proving exponential convergence for analytic eigenfunctions and demonstrating numerically that hp-refinements yield exponential convergence in terms of the cubic root of degrees of freedom for eigenfunctions with finite Sobolev regularity.
We discuss the $p$- and the $hp$-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. We present in details 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.