|Abstract:|| The Virtual Element Method is a generalization of the classical Finite Element Method to arbitrary element-geometry. In the first part of the talk we present a new family of Virtual Elements for the Stokes problem on polygonal meshes. By a proper choice of the Virtual space of velocities and the associated degrees of freedom, we can guarantee that the final discrete velocity is pointwise divergence-free, and not only in a relaxed (projected) sense, as it happens for more standard elements. Moreover, we show that the discrete problem is immediately equivalent to a reduced
problem with fewer degrees of freedom, thus yielding a very efficient scheme. The focus of the second part of the talk is on developing a Virtual Element Method for Darcy and Brinkman equations. We use introduce a slightly different Virtual Element space having two fundamental properties: the L2-projection onto the space of polynomials of degree k
(being k the order of the method) is exactly computable on the basis of the degrees of freedom, and the associated discrete kernel is still pointwise divergence-free. The resulting numerical scheme for the Darcy problem has optimal order of convergence and H1 conforming velocity solution. In particular we obtain a Virtual Element scheme that is accurate for both Darcy and Stokes equations. Then we can apply the same approach to develop a stable virtual element method for Brinkman equations (that is combination of Stokes and Darcy equations). We provide a rigorous error analysis of the method and several numerical tests, including a
comparison with a different Virtual Element choice.