Basic principles of hp Virtual Elements on quasiuniform meshes

In the present paper we initiate the study of $hp$ Virtual Elements. We focus on the case with uniform polynomial degree across the mesh and derive theoretical convergence estimates that are explicit both in the mesh size $h$ and in the polynomial degree $p$ in the case of finite Sobolev regularity. Exponential convergence is proved in the case of analytic solutions. The theoretical convergence results are validated in numerical experiments. Finally, an initial study on the possible choice of local basis functions is included

THE HP VERSION OF THE VIRTUAL ELEMENT METHOD

The interest in Galerkin methods for the approximation of partial differential equations on polytopal meshes has recently grown. The virtual element method (VEM) is one of the most successful approaches enabling computation on such meshes. So far, only the h version of the method has been investigated; here, the convergence of the error is obtained by keeping fixed the dimension of local spaces, while refining the mesh. Contrarily, the p version of a Galerkin method consists in achieving convergence by keeping fixed the decomposition of the domain and increasing the dimension of local spaces. The combination of the h and p versions goes under the name of hp version of the method under consideration. The present thesis aims to dovetail the technology of VEM with the p and the hp refinement strategies. Particular emphasis is offered to the approximation of Poisson and Laplace problems, a priori and a posteriori error analysis, multigrid solvers, stabilization of the method and ill-conditioning of the stiffness matrix