Non-conforming harmonic virtual element method: $h$- and $p$-versions
Provides a more efficient numerical method for solving elliptic PDEs on polygonal domains, with theoretical tools applicable to broader classes of non-conforming methods.
The paper develops and analyzes non-conforming harmonic virtual element methods for the Dirichlet-Laplace problem, achieving faster convergence per degree of freedom than standard VEM and demonstrating exponential convergence in hp-version tests.
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.