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Error analysis of some finite element methods for the Stokes problem. (English) Zbl 0702.65095
The author proves the optimal convergence for some two-dimensional finite element methods for the Stokes equations. First two families of ‘Taylor- Hood’ type methods: the triangular \(P_ 3-P_ 2\) element and the \(Q_ k-Q_{k-1}\), \(k\geq 2\), family of quadrilateral elements are considered. Finally two new low-order methods with piecewise constant approximations for the pressure are introduced and analyzed. Macro-element technique, introduced by the author [ibid. 42, 9-23 (1984; Zbl 0535.76037)] is used in a slightly more practical form for error analysis.
It is claimed that the results of the paper are trivially also valid when the same finite element spaces are used for equations of (nearly) incompressible elasticity. The author admits that some of the results presented herein have also been obtained by F. Brezzi and R. S. Falk [Stability of a higher order Hood-Taylor method, SIAM, J. Numer. Anal. (to appear)].
Reviewer: H.K.Verma

MSC:
65Z05 Applications to the sciences
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
35Q30 Navier-Stokes equations
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