## Preprint 2009-02

## Gabriel N. Gatica, Luis F. Gatica, Antonio Marquez:

### Augmented mixed finite element methods for a curl-based formulation of the two-dimensional Stokes problem

### Abstract:

In this paper we consider an augmented curl-based mixed formulation of the Stokes problem in the plane, and then introduce and analyze stable mixed finite element methods to solve the associated Galerkin scheme. In this way, we further extend similar procedures applied recently to linear elasticity and to other mixed formulations for incompressible fluid flows. Indeed, our approach is based on the introduction of the Galerkin least-squares type terms arising from the corresponding constitutive and equilibrium equations, and from the Dirichlet boundary condition for the velocity, all of them multiplied by stabilization parameters. Then, we show that these parameters can be suitably chosen so that the resulting operator equation induces a strongly coercive bilinear form, whence the associated Galerkin scheme becomes well posed for any choice of finite element subspaces. In particular, we can use continuous piecewise linear velocities, piecewise constant pressures, and rotated Raviart-Thomas elements for the stresses. Next, we derive reliable and efficient residual-based a posteriori error estimators for the augmented mixed finite element schemes. In addition, 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. The present work should be considered as a first step aiming finally to derive augmented mixed finite element methods for curl-based formulations of the three-dimensional Stokes problem.

This preprint gave rise to the following definitive publication(s):

**Gabriel N. GATICA, Luis F. GATICA, Antonio MARQUEZ: ***Augmented mixed finite element methods for a vorticity-based velocity-pressure-stress formulation of the Stokes problem in 2D*. International Journal for Numerical Methods in Fluids, vol. 67, 4, pp. 450-477, (2011).