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Analysis of a streamline diffusion finite element method for the Stokes and Navier-Stokes equations. (English) Zbl 0843.76052

Summary: For the Stokes equations with convection and the incompressible Navier-Stokes equations, we analyze a streamline diffusion finite element method that is capable of balancing both the convection and the pressure, thus allowing the use of arbitrary pairs of velocity-pressure spaces. For the linear problem, we obtain for all mesh-Péclet numbers optimal error estimates in natural norms including the \(L^2\)-norm of the pressure. The same holds for the nonlinear problem, which close to a regular branch of solutions, i.e., the linearized operator, is an isomorphism, the norm of the inverse of which still depends on the Reynolds number. Consequently, the dependence of the error constants on the Reynolds number is not completely resolved in this case.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76D07 Stokes and related (Oseen, etc.) flows
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