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Virtual Element Methods for general second order elliptic problems on polygonal meshes

We consider the discretization of a boundary value problem for a general linear second-order elliptic operator with smooth coefficients using the Virtual Element approach. As in [59] the problem is supposed to have a unique solution, but the associated bilinear form is not supposed to be coercive. Contrary to what was previously done for Virtual Element Methods (as for instance in [9]), we use here, in a systematic way, the L^2-projection operators as designed in [1]. In particular, the present method does not reduce to the original Virtual Element Method of [9] for simpler problems as the classical Laplace operator (apart from the lowest order cases). Numerical experiments show the accuracy and the robustness of the method, and they show as well that a simple-minded extension of the method in [9] to the case of variable coefficients produces, in general, sub-optimal results.