Non-conforming Harmonic Virtual Element Method: $$h$$h- and $$p$$p-Versions

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.