  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.

Numerical Analysis