Augmented Mixed Finite Element Methods for the Stationary Stokes Equations

Abstract. In this paper we introduce and analyze two augmented mixed finite element methods for a velocity-pressure-stress formulation of the stationary Stokes equations. Our approach, which extends analogue results for linear elasticity problems, is based on the introduction of the Galerkin least-squares type terms arising from the constitutive and equilibrium equations, and the Dirichlet boundary condition for the velocity, all them multiplied by suitable stabilization parameters. We show that these parameters can be chosen so that the resulting augmented variational formulations are defined by strongly coercive bilinear forms, whence the associated Galerkin schemes become well posed for any choice of finite element subspaces. In particular, we can use continuous piecewise linear velocities, piecewise constant pressures, and Raviart-Thomas elements for the stresses, thus yielding a number of unknowns behaving asymptotically as 5 times the number of triangles of the triangulation. Alternatively, the above factor reduces to 4 when a second augmented variational formulation, involving only the velocity and the stress as unknowns, is employed. Next, we derive a reliable and efficient residual-based a posteriori error estimator for the augmented mixed finite element schemes. Finally, several numerical experiments illustrating the performance of the augmented mixed finite element methods, confirming the properties of the a posteriori estimators, and showing the behaviour of the associated adaptive algorithms are reported.