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

by   Silvia Bertoluzza, et al.

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 Ω_h of the domain Ω, 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 Ω_h. Numerical experiments validate the theory.



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