Weakly imposed Dirichlet boundary conditions for 2D and 3D Virtual Elements

In the framework of virtual element discretizazions, we address the problem of imposing non homogeneous Dirichlet boundary conditions in a weak form, both on polygonal/polyhedral domains and on two/three dimensional domains with curved boundaries. We consider a Nitsche's type method [43,41], and the stabilized formulation of the Lagrange multiplier method proposed by Barbosa and Hughes in [9]. We prove that also for the virtual element method (VEM), provided the stabilization parameter is suitably chosen (large enough for Nitsche's method and small enough for the Barbosa-Hughes Lagrange multiplier method), the resulting discrete problem is well posed, and yields convergence with optimal order on polygonal/polyhedral domains. On smooth two/three dimensional domains, we combine both methods with a projection approach similar to the one of [31]. We prove that, given a polygonal/polyhedral approximation $\Omega_h$ of the domain $\Omega$, an optimal convergence rate can be achieved by using a suitable correction depending on high order derivatives of the discrete solution along outward directions (not necessarily orthogonal) at the boundary facets of $\Omega_h$. Numerical experiments validate the theory.