Thesis Summary:

We consider a non-standard mixed approach for the Stokes problem in which the velocity, the pressure, and the pseudostress are the main unknowns. Alternatively, the pressure can be eliminated from the original equations, thus yielding an equivalent formulation with only two unknowns. In this work we develop a priori and a posteriori error analyses of both approaches. In fact, we first apply the Babuska-Brezzi theory to prove the well-posedness of the continuous and discrete formulations. In particular, we show that Raviart-Thomas elements of order k ge 0 for the pseudostresses, and piecewise polynomials of degree k for the velocities and the pressures, ensure unique solvability and stability of the associated Galerkin schemes. Then, we derive a reliable and eficient residual-based a posteriori error estimator. Finally, we provide several numerical results illustrating the good performance of the mixed finite element method and confirming the theoretical properties of the estimator.

ISI Publications from the Thesis

Gabriel N. GATICA, Antonio MARQUEZ, Manuel A. SANCHEZ: Pseudostress-based mixed finite element methods for the Stokes problem in $R^n$ with Dirichlet boundary conditions. I: A priori error analysis. Communications in Computational Physics, vol. 12, 1, pp. 109-134, (2012).

Gabriel N. GATICA, Antonio MARQUEZ, Manuel A. SANCHEZ: A priori and a posteriori error analyses of a velocity-pseudostress formulation for a class of quasi-Newtonian Stokes flows. Computer Methods in Applied Mechanics and Engineering, vol. 200, 17-20, pp. 1619-1636, (2011).

Gabriel N. GATICA, Antonio MARQUEZ, Manuel A. SANCHEZ: Analysis of a velocity-pressure-pseudostress formulation for the stationary Stokes equations. Computer Methods in Applied Mechanics and Engineering, vol. 199, 17-20, pp. 1064-1079, (2010)