High order VEM on curved domains

Abstract

We deal with the virtual element method (VEM) for solving the Poisson equation on a domain $\Omega$ with curved boundary. Given a polygonal approximation ${\Omega }_h$ of the domain $\Omega$, the standard order $m$ VEM [6], for $m$ increasing, leads to a suboptimal convergence rate. We adapt the approach of [14] 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 ${\Omega }_h$, which, to retain computability, is evaluated after applying the projector ${\Pi }^{\nabla }$ onto the space of polynomials. Numerical experiments confirm the theory.