Multilevel algorithms for mixed problems. II: Treatment of the mini- element. (English) Zbl 0669.65083

Verfürth, R.

SIAM J. Numer. Anal. 25, No. 2, 285-293 (1988).

The author presents a multilevel algorithm for the mixed finite element approximation of the Stokes problem by the mini-element. The mini-element does not fit into the abstract framework of part I [ibid. 21, 264-271 (1984; Zbl 0534.65065)] because the bubble functions, essential for the stability of the mini-element, are not local when passing from a coarse to fine grid. A theory is developed based on the strengthened Cauchy inequality to bound this effect.

Reviewer: J.Mandel

MSC:

65N22	Numerical solution of discretized equations for boundary value problems involving PDEs
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
76D05	Navier-Stokes equations for incompressible viscous fluids
65F10	Iterative numerical methods for linear systems
35Q30	Navier-Stokes equations
76D07	Stokes and related (Oseen, etc.) flows

Keywords:

convergence; multilevel algorithm; finite element; Stokes problem; mini- element; stability; Cauchy inequality

Citations:

Zbl 0534.65065