Commun. Comput. Phys., 12 (2012), pp. 109-134.


Pseudostress-Based Mixed Finite Element Methods for the Stokes Problem in R^n with Dirichlet Boundary Conditions. I: A Priori Error Analysis

Gabriel N. Gatica 1*, Antonio Marquez 2, Manuel A. Sanchez 1

1 CI^2MA and Departamento de Ingenieria Matematica, Universidad de Concepcion, Casilla 160-C, Concepcion, Chile.
2 Departamento de Construccion e Ingenieria de Fabricacion, Universidad de Oviedo, Oviedo, Espana.

Received 1 March 2011; Accepted (in revised version) 4 October 2011
Available online 20 January 2012
doi:10.4208/cicp.010311.041011a

Abstract

We consider a non-standard mixed method for the Stokes problem in R^n, $n\in\{2,3\}$, with Dirichlet boundary conditions, in which, after using the incompressibility condition to eliminate the pressure, the pseudostress tensor $\sigma$ and the velocity vector $\u$ become the only unknowns. Then, we apply the Babuska-Brezzi theory to prove the well-posedness of the corresponding continuous and discrete formulations. In particular, we show that Raviart-Thomas elements of order $k \ge 0$ for $\sigma$ and piecewise polynomials of degree k for u ensure unique solvability and stability of the associated Galerkin scheme. In addition, we introduce and analyze an augmented approach for our pseudostress-velocity formulation. The methodology employed 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 of them multiplied by suitable stabilization parameters. We show that these parameters can be chosen so that the resulting augmented variational formulation is defined by a strongly coercive bilinear form, whence the associated Galerkin scheme becomes well posed for any choice of finite element subspaces. For instance, Raviart-Thomas elements of order $k \ge 0$ for $\sigma$ and continuous piecewise polynomials of degree k+1 for u become a feasible choice in this case. Finally, extensive numerical experiments illustrating the good performance of the methods and comparing them with other procedures available in the literature, are provided.

AMS subject classifications: 65N15, 65N30, 65N50, 74B05

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Key words: Mixed finite element, pseudostress, incompressible flow.

*Corresponding author.
Email: ggatica@ing-mat.udec.cl (G. N. Gatica), amarquez@uniovi.es (A. Marquez), manusanchez@udec.cl (M. A. Sanchez)
 

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