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

In the framework of virtual element discretizations, 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 (Nitsche et al. 1970; Juntunen et al. 2009) and the stabilized formulation of the Lagrange multiplier method proposed by Barbosa and Hughes in Barbosa and Hughes (1991) . 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 Bramble et al. (1972). We prove that, given a polygonal/polyhedral approximation ohm h of the domain ohm, 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.