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Space and time error estimates for a first-order, pressure-stabilized finite element method for the incompressible Navier-Stokes equations. (English) Zbl 1011.76041
Summary: We analyze a pressure-stabilized finite element method for unsteady incompressible Navier-Stokes equations in primitive variables, for the time discretization we focus on a fully implicit monolithic scheme. We provide some error estimates for fully discrete solution which show that the velocity is first-order accurate in time step and attains optimal order accuracy in the mesh size for a given spatial interpolation, both in the spaces ${L}^{2}\left({\Omega }\right)$ and ${H}_{0}^{1}\left({\Omega }\right)$; the pressure solution is shown to be of order $\frac{1}{2}$ accurate in time step and also optimal in mesh size. These estimate are proved assuming only a weak compatibility condition on approximating spaces of velocity and pressure, which is satisfied by equal-order interpolations.
##### MSC:
 76M10 Finite element methods (fluid mechanics) 76D05 Navier-Stokes equations (fluid dynamics) 65M15 Error bounds (IVP of PDE)