Non-conforming harmonic virtual element method: $h$- and $p$-versions

Abstract

We study the $h$- and $p$-versions of non-conforming harmonic virtual element methods (VEM) for the approximation of the Dirichlet-Laplace problem on a 2D polygonal domain, providing quasi-optimal error bounds. Harmonic VEM do not make use of internal degrees of freedom. This leads to a faster convergence, in terms of the number of degrees of freedom, as compared to standard VEM. Importantly, the technical tools used in our $p$-analysis can be employed as well in the analysis of more general non-conforming finite element methods and VEM. The theoretical results are validated in a series of numerical experiments. The $hp$-version of the method is numerically tested, demonstrating exponential convergence with rate given by the square root of the number of degrees of freedom.

References in corpus (3)

• A Virtual Element Method for elastic and inelastic problems on polytope meshes
• BDDC and FETI-DP for the Virtual Element Method
• Exploring High-order three dimensional Virtual Elements: bases and stabilizations

Cited by in corpus (4)

• A three-dimensional Hellinger-Reissner Virtual Element Method for linear elasticity problems
• Virtual Element for the Buckling Problem of Kirchhoff-Love plates
• Enrichment of the nonconforming virtual element method with singular functions
• The virtual element method for eigenvalue problems with potential terms on polytopal meshes