Exponential convergence of the hp Virtual Element Method with corner singularities

Abstract

In the present work, we analyze the $hp$ version of Virtual Element methods for the 2D Poisson problem. We prove exponential convergence of the energy error employing sequences of polygonal meshes geometrically refined, thus extending the classical choices for the decomposition in the $hp$ Finite Element framework to very general decomposition of the domain. A new stabilization for the discrete bilinear form with explicit bounds in $h$ and $p$ is introduced. Numerical experiments validate the theoretical results. We also exhibit a numerical comparison between $hp$ Virtual Elements and $hp$ Finite Elements.

30 pages, 8 figures

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• A Virtual Element Method for elastic and inelastic problems on polytope meshes
• Divergence free Virtual Elements for the Stokes problem on polygonal meshes
• Virtual Element Method for geomechanics on reservoir grids
• Serendipity Nodal VEM spaces
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Cited by in corpus (4)

• BDDC and FETI-DP for the Virtual Element Method
• Weakly imposed Dirichlet boundary conditions for 2D and 3D Virtual Elements
• High-order Virtual Element Method on polyhedral meshes
• Algebraic Multigrid Methods For Virtual Element Discretizations: A Numerical Study