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A least squares Galerkin-Petrov nonconforming mixed finite element method for the stationary conduction-convection problem. (English) Zbl 1189.76345
The authors consider a steady-state conduction-convection problem in two space dimensions, which corresponds to incompressible Navier-Stokes equations in Boussinesq approximation. The set of partial differential equations is discretized employing a least square formulation together with a Petrov-Galerkin method which uses non-conforming finite elements on triangular meshes. Different polynomial degrees are used for velocity, temperature and pressure. For the latter linear elements are applied, while for the two former variables quadratic ones are utilized. Existence and uniqueness of the discrete solution are proven together with error estimates which are optimal, supposing sufficiently large viscosity. Furthermore, it is shown that the mixed finite element spaces do not need to satisfy the inf-sup condition.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76R10 Free convection
80A20 Heat and mass transfer, heat flow (MSC2010)
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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