A virtual element method with arbitrary regularity

We develop and analyse a new family of virtual element methods on unstructured polygonal meshes for the diffusion problem in primal form, which uses arbitrarily regular discrete spaces. The degrees of freedom are (a) solution and derivative values of various degrees at suitable nodes and (b) solution moments inside polygons. The convergence of the method is proved theoretically and an optimal error estimate is derived. Numerical experiments confirm the convergence rate that is expected from the theory.