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## Sergio Caucao, David Mora, Ricardo Oyarzua:

### Abstract:

We propose and analyse a mixed finite element method for the nonstandard pseudostress-velocity formulation of the Stokes problem with varying density  in R$^$d, d \in {2,3}. Since the resulting variational formulation does not have the standard dual-mixed structure, we reformulate the continuous problem as an equivalent fixed-point problem. Then, we apply the classical Babuska-Brezzi theory to prove that the associated mapping T is well de fined, and assuming that \|\frac{\nabla \rho}{\rho}\| is suficiently small, we show that T is a contraction mapping, which implies that the variational formulation is well-posed. Under the same hypothesis on \rho we prove stability of the continuous problem. Next, adapting to the discrete case the arguments of the continuous analysis, we are able to establish suitable hypotheses on the fi nite element subspaces ensuring that the associated Galerkin scheme becomes well-posed. A feasible choice of subspaces is given by Raviart-Thomas elements of order k \ge 0 for the pseudostress and polynomials of degree k for the velocity. Finally, several numerical results illustrating the good performance of the method with these discrete spaces, and con firming the theoretical rate of convergence, are provided.

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Esta prepublicacion dio origen a la(s) siguiente(s) publicación(es) definitiva(s):

Sergio CAUCAO, David MORA, Ricardo OYARZUA: A priori and a posteriori error analysis of a pseudostress-based mixed formulation of the Stokes problem with varying density. IMA Journal of Numerical Analysis, vol. 36, 2, pp. 947-983, (2016).