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The Virtual Element Method (VEM) is a new family of numerical methods for theapproximation of partial differential equations, where the geometry of thepolytopal mesh elements can be very general. The aim of this article is toextend the balancing domain decomposition by constraints (BDDC) preconditionerto the solution of the saddle-point linear system arising from a VEMdiscretization of the two-dimensional Stokes equations. Under suitablehypotesis on the choice of the primal unknowns, the preconditioned linearsystem results symmetric and positive definite, thus the preconditionedconjugate gradient method can be used for its solution. We provide atheoretical convergence analysis estimating the condition number of thepreconditioned linear system. Several numerical experiments validate thetheoretical estimates, showing the scalability and quasi-optimality of themethod proposed. Moreover, the solver exhibits a robust behavior with respectto the shape of the polygonal mesh elements. We also show that a fasterconvergence could be achieved with an easy to implement coarse space, slightlylarger than the minimal one covered by the theory.

BDDC preconditioners for divergence free virtual element discretizations of the Stokes equations