Abstract. We introduce the nonconforming Virtual Element Method (VEM) for the approximation of second order elliptic problems. We present the construction of the new element in two and three dimensions, highlighting the main differences with the conforming VEM and the classical nonconforming finite element methods. We provide the error analysis and establish the equivalence with a family of mimetic finite difference methods.

We propose a family of preconditioners for linear systems of equations arising from a piecewise polynomial symmetric Interior Penalty Discontinuous Galerkin (IP-DG) discretization of H(curl, Ω)elliptic boundary value problems on conforming meshes. The design and analysis of the proposed preconditioners relies on the auxiliary space method (ASM) employing an auxiliary space of H(curl, Ω)conforming finite element functions together with a relaxation technique (local smoothing). On simplicial meshes, the proposed preconditioner enjoys asymptotic optimality with respect to mesh refinement. It is also robust with respect to jumps in the coefficients ν and β in the second and zeroth order parts of the operator, respectively, except when both coefficients are discontinuous and the problem is curldominated in some regions and reaction dominated in others. On quadrilateral/hexahedral meshes our ASM may fail, since the related H(curl, Ω)-conforming finite element space does not provide a spectrally accurate discretization.