The harmonic virtual element method: stabilization and exponential convergence for the Laplace problem on polygonal domains

We introduce the harmonic virtual element method (harmonic VEM), a modification of the virtual element method (VEM) for the approximation of the 2D Laplace equation using polygonal meshes. The main difference between the harmonic VEM and the VEM is that in the former method only boundary degrees of freedom are employed. Such degrees of freedom suffice for the construction of a proper energy projector on (piecewise harmonic) polynomial spaces. The harmonic VEM can also be regarded as an "$H^1$-conformisation" of the Trefftz discontinuous Galerkin-finite element method (TDG-FEM). We address the stabilization of the proposed method and develop an $hp$ version of harmonic VEM for the Laplace equation on polygonal domains. As in Trefftz DG-FEM, the asymptotic convergence rate of harmonic VEM is exponential and reaches order $\mathcal O ( \exp(-b\sqrt[2]{N}))$, where $N$ is the number of degrees of freedom. This result overperformes its counterparts in the framework of $hp$ FEM and $hp$ VEM, where the asymptotic rate of convergence is of order $\mathcal O ( \exp(-b\sqrt[3]{N}) )$.