This article is concerned with a dual mixed formulation of the Navier-Stokes system in a polygonal domain of the plane with Dirichlet boundary conditions and its numerical approximation. The gradient tensor, a quantity of practical interest, is introduced as a new unknown. The problem is then approximated by a mixed finite element method. Quasi-optimal a priori error estimates are obtained. These a priori error estimates, an abstract nonlinear theory (similar to (Verfürth, RAIRO Model Math Anal Numer 32 (1998), 817-842)) and a posteriori estimates for the Stokes system from (Farhloul et al., Numer Funct Anal Optim 27 (2006), 831-846) lead to an a posteriori error estimate for the Navier-Stokes system.
Abstract. This paper is concerned with the mixed formulation of the Boussinesq equations in two-dimensional domains and its numerical approximation. The paper deals first with existence and uniqueness results, as well as the description of the regularity of any solution. The problem is then approximated by a mixed finite element method, where the gradient of the velocity and the gradient of the temperature, quantities of practical importance, are introduced as new unknowns. An existence result for the finite element solution and convergence results are proved near a nonsingular solution. Quasi-optimal error estimates are finally presented.
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