A Cascade Algorithm for the Stokes Equations Dietrich Braess Wolfgang Dahmen Abstract A variant of multigrid schemes for the Stokes problem is discussed. In particular, we propose and analyse a cascadic version for the Stokes problem. The analysis of the transfer between the grids requires special care in order to establish that the complexity is the same as that for classical multigrid algorithms. Key words: Saddle point problems, Stokes problem, cascadic iteration, smoother, projection method 1 Introduction Multigrid methods without coarse grid corrections have been defined and applied to elliptic problems of second order by Bornemann and Deuflhard [1]. They have called it a cascadic algorithm and showed that an optimal iteration method with respect to the energy norm is obtained if conforming elements are used. Deuflhard’s starting point for the cascadic multigrid method was the idea that it should be sufficient to start the iteration at the level i with a good approximation from the level i − 1. A similar idea can already be found in Chapter 9 of Wachspress’ book [10] from 1966, i.e. from the period in which also the first theoretical investigations of multigrid methods were made. Later Shaidurov [7] established in essence a recursion relation of the form ∥u i − v i ∥ 1 ≤∥u i −1 − v i −1 ∥ 1 + c h i m i (1.1) for some finite element problems with full regularity. Here u i denotes the exact solution on the level i and v i its approximation computed after m i steps. The accumulation of the error is no problem since the iteration steps on the lower levels are cheap. It is crucial for the optimality of the algorithm that the error from the previous level enters with a factor of precisely 1. Since it was not clear whether a constant factor greater than 1 is encountered in the transfer for nonconforming elements, there are no serious conjectures for the latter families. There is another difference to classical multigrid algorithms. The recursion relation (1.1) refers only to the energy norm, and it has been proved in [2] that the cascadic version is in general not optimal for the L 2 -norm. This is in contrast to classical multigrid algorithms, see [6, 11], where one can more easily move between the H 1 -norm and the L 2 -norm. 1