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We deal with the virtual element method (VEM) for solving the Poisson equation on a domain with curved boundary. Given a polygonal approximation of the domain , the standard order VEM , for increasing, leads to a suboptimal convergence rate. We adapt the approach of  to VEM and we prove that an optimal convergence rate can be achieved by using a suitable correction depending on high order normal derivatives of the discrete solution at the boundary edges of , which, to retain computability, is evaluated after applying the projector onto the space of polynomials. Numerical experiments confirm the theory.
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Silvia Bertoluzza, Micol Pennacchio, Daniele Prada, High order VEM on curved domains. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 30 (2019), no. 2, pp. 391–412