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We introduce the harmonic virtual element method (harmonic VEM), amodification of the virtual element method (VEM) for the approximation of the2D Laplace equation using polygonal meshes. The main difference between theharmonic VEM and the VEM is that in the former method only boundary degrees offreedom are employed. Such degrees of freedom suffice for the construction of aproper energy projector on (piecewise harmonic) polynomial spaces. The harmonicVEM can also be regarded as an "$H^1$-conformisation" of the Trefftzdiscontinuous Galerkin-finite element method (TDG-FEM). We address thestabilization of the proposed method and develop an $hp$ version of harmonicVEM for the Laplace equation on polygonal domains. As in Trefftz DG-FEM, theasymptotic convergence rate of harmonic VEM is exponential and reaches order$\mathcal O ( \exp(-b\sqrt[2]{N}))$, where $N$ is the number of degrees offreedom. This result overperformes its counterparts in the framework of $hp$FEM and $hp$ VEM, where the asymptotic rate of convergence is of order$\mathcal O ( \exp(-b\sqrt[3]{N}) )$.

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