## Preprint 2013-20

## Sergio Caucao, David Mora, Ricardo Oyarzua:

### Analysis of a mixed-FEM for the pseudostress-velocity formulation of the Stokes problem with varying density

### 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 Babuska-Brezzi theory to prove that the associated mapping T is well defined, and assuming that \|\frac{\nabla \rho}{\rho}\| is suficiently 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 finite 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 confirming the theoretical rate of convergence, are provided.

This preprint gave rise to the following definitive publication(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).