Thesis Summary:

In this work we develop a discretization using virtual elements for the Brinkman problem formulated in terms of the current function (also called the flow function) of the velocity field. We write a variational formulation and propose a discretization using virtual elements of class C1 of arbitrary order k≥2. The speed is obtained through a post-process of the current function. Under standard assumptions of the computational domain, we test error estimates for the current function. We also establish error estimates in the L2 and H1 standards through arguments of duality classics. For k = 3, we propose a strategy to approximate the pressure of the fluid, through a generalized Poisson problem with data from the current function, which is based on a discrete formulation with virtual elements of class C0. In addition, under the hypothesis of mesh quasi-uniformity, error estimates are established in the H1 norm for pressure. Finally, some numerical results are reported using different families of polygonal meshes, which illustrate the good behavior of the discrete scheme and corroborate our theoretical results.