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Analysis of a pressure-stabilized finite element approximation of the stationary Navier-Stokes equations. (English) Zbl 0988.76049
Summary: We analyze a finite element approximation of stationary Navier-Stokes equations that allows the use of equal velocity-pressure interpolation. The idea is to introduce as unknown of the discrete problem the projection of pressure gradient (multiplied by suitable algorithmic parameters) onto the space of continuous vector fields. The difference between these two vectors (pressure gradient and projection) is also introduced in the continuity equation. The resulting formulation is shown to be stable and optimally convergent, both in a norm associated to the problem and in ${L}^{2}$ norm for both velocities and pressure. This is proved first for the linearized Stokes problem, and then is extended to the nonlinear case. The analysis relies on an inf-sup condition that is much weaker than for the standard Galerkin approximation, in spite of the fact that the present method is only a minor modification of this.
##### MSC:
 76M10 Finite element methods (fluid mechanics) 76D05 Navier-Stokes equations (fluid dynamics) 65N12 Stability and convergence of numerical methods (BVP of PDE)