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A stable nonconforming quadrilateral finite element method for the stationary Stokes and Navier-Stokes equations. (English) Zbl 0947.76047
Summary: Recently, J. Douglas jun., J. E. Santos, D. Sheen and X. Ye [M2AN, Math. Model. Numer. Anal. 33, No. 4, 747-770 (1999; Zbl 0941.65115)] introduced a new, low-order, non-conforming rectangular element for scalar elliptic equations. Here we apply this element in the approximation of each velocity component for stationary Stokes and Navier-Stokes equations, along with a piecewise-constant element for pressure. We obtain a stable element in both cases in which optimal error estimates for the approximation of both velocity and pressure in $$L^2$$ can be established, as well as in the case when the optimal error estimate can be established in a broken $$H^1$$-norm for the velocity.

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