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An efficient smoother for the Stokes problem. (English) Zbl 0874.65095
The goal of this interesting paper is to design the smoothing procedure in the multigrid algorithms for the solution of the Stokes and the Navier-Stokes equations. The construction is based on the SIMPLE method which was introduced by S. V. Patankar and D. B. Spalding [Internat. J. Heat Mass Transfer 15, 1787-1806 (1972; Zbl 0246.76080)] and which is often used in numerical fluid mechanics.
The authors present a smoother which is obtained from a variant of the pressure correction steps which is the second iteration step in the SIMPLE method. Applied to variational problems without restrictions, the presented approach leads to the classical multigrid procedure with smoothing by the Jacobi or Gauss-Seidel iteration. A convergence rate $$O(m^{-1})$$ is obtained where $$m$$ denotes the number of smoothing steps. Moreover, the differences among the presented algorithm, the SIMPLE one and the SIMPLEC method of J. P. van Doormaal and G. D. Raithby [Numer. Heat Transfer 7, 147-163 (1984, Zbl 0553.76005)] are discussed.
Numerical results for finite element discretizations (with three different grids) of the 2D Stokes problem on a square are presented. Examples of computational test problems are also given.

##### MSC:
 65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs 65F10 Iterative numerical methods for linear systems 35Q30 Navier-Stokes equations 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
Wesseling
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