A priori and a posteriori error analysis of a mixed scheme for the Brinkman problem

This paper deals with the analysis of new mixed finite element methods for the Brinkman equations formulated in terms of velocity, vorticity and pressure. Employing the Babuška–Brezzi theory, it is proved that the resulting continuous and discrete variational formulations are well-posed. In particular, we show that Raviart– Thomas elements of order k ≥ 0 for the approximation of the velocity field, piecewise continuous polynomials of degree k + 1 for the vorticity, and piecewise polynomials of degree k for the pressure, yield unique solvability of the discrete problem. On the other hand, we also show that families of finite elements based on Brezzi–Douglas– Marini elements of order k+1 for the approximation of velocity, piecewise continuous polynomials of degree k+2 for the vorticity, and piecewise polynomials of degree k for the pressure ensure thewell-posedness of the associatedGalerkin scheme.We note that these families provide exactly divergence-free approximations of the velocity field. We establish a priori error estimates in the natural norms with constants independent B Ricardo Ruiz-Baier [email protected] Verónica Anaya [email protected] David Mora [email protected] Ricardo Oyarzúa [email protected] 1 GIMNAP, Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Casilla 5-C, Concepción, Chile 2 Centro de Investigación en Ingeniería Matemática (CI2MA), Universidad de Concepción, Casilla 160-C, Concepción, Chile 3 Institut des Sciences de la Terre, University of Lausanne, Géopolis Unil-Mouline, 1015 Lausanne, Switzerland