An New Improved Uzawa Method for Finite Element Solution of Stokes Problem

Tchebychev iteration may be used for acceleration convergence of an iterative algorithm to solve a general linear system equation. Associating it with the Uzawa method, we suggest a new iterative solution method for the Stokes problems. The new algorithm retains the simplicity and robustness of the Uzawa method. So it requires almost no additional cost of computation, in terms of storage or CPU time, yet it provides the property of speed up convergence. Numerical tests showed that the algorithm of this type have much faster convergence rates than both the original Uzawa iterative algorithm and the augmented Lagrangian method. 1 Introduction Numerical solution of the Stokes problem is a basic problem for computation of the incompressible ¯ows of a viscous ¯uid and the stress-displacements in an incom-pressible elasticity. There are many ways to solve the Stokes problem, such as Uzawa-like [4±10, 16, 25], SIMPLE like [14] and multigrid-like methods [5, 23]. In this work we only consider the Uzawa-like methods. The ®rst version of the Uzawa algorithm was given by the group of Uzawa in 1958. It is a direct application of the gradient method to the minimization problem of the dual functional of the Stokes problem and therefore is an approach of iterative solution of the Stokes problem. The most attractive character of the Uzawa algorithm is its extreme simplicity and robustness. However, unfortunately, its speed of convergence is slow in some cases and therefore it requires too many iterations to obtain suf®cient accuracy. To overcome this major shortage of the Uzawa algorithm , many researchers have suggested various improved versions of the Uzawa algorithm, for example, augmented Lagrangian [4, 7, 8], multigrid [5, 21] and preconditioning methods [4±6, 10, 17]. Some of them increase the cost of the computation per iteration, in terms of storage or CPU time, though they speed up the convergence of the Uzawa algorithm. However, the augmented Lagrangian method proposed by Fortin and Glowinski in 1983 does not almost require additional effort of computation compared with the original Uzawa algorithm and also can raise the rate of convergence of the Uzawa algorithm. It uses an augmented Lagrangian formulation of the Stokes problem. Through applying the gradient method to this augmented Lagrangian minimization problem, the improved Uzawa algorithm, augmented Lagrangian method, is available. The augmented Lagrangian method is of the simple form like the Uzawa algorithm and therefore is widely adopted in calculating the Stokes …