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A multigrid solver for a stabilized finite element discretization of the Stokes problems. (English) Zbl 0616.65104

Multigrid methods II, Proc. 2nd Eur. Conf., Cologne/Ger. 1985, Lect. Notes Math. 1228, 1-6 (1986).
[For the entire collection see Zbl 0596.00016.]
For solving finite element discretized approximations to a variational formulation of the Stokes equations, the corresponding spaces of piecewise linear functions on the triangular meshes provide an instable finite element method scheme. F. Brezzi and J. Pitk√§ranta [Notes Numer. Fluid Mech. 10, 11-19 (1984; Zbl 0552.76002)] proposed two ways of stabilization at little cost: a) by adding ”bubble functions”; b) by modifying the discrete equations.
We have tried the first way, and more recently the second to which we restrict ourselves in this paper. We give a multigrid solver for the Stokes equations which is very efficient, at least for the regular mesh of our first experiments.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
35J25 Boundary value problems for second-order elliptic equations
76D07 Stokes and related (Oseen, etc.) flows