An new improved Uzawa method for finite element solution of Stokes problem

Abstract

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.

Numerical solution of saddle point problems

“…An Uzawatype method with variable relaxation parameters was proposed by Hu and Zou (2001). Uzawa's method is still being actively developed by many researchers: recent papers discussing various extensions and improvements of Uzawa's classical algorithm include Bertrand and Tanguy (2002), Bramble, Pasciak and Vassilev (2000), Cao (2004b), Chen (1998), Cui (2004), Hu and Zou (2002), Liu and Xu (2001), Maday, Meiron, Patera and Ronquist (1993), Nochetto and Pyo (2004), Zsaki, Rixen and Paraschivoiu (2003). Not all applications of Uzawa's method are to fluid flow problems: see Ito and Kunisch (1999) for a recent application to image restoration.…”

Compression moulding of SMC: In situ experiments, modelling and simulation

“…In the case of a linear system (n = m = 1), the problem can be solved by applying the Uzawa algorithm for decoupling the momentum equations and the incompressibility constraint. To reduce the number of iterations up to convergence, the Uzawa algorithm was modified by solving the saddle-point problem for the augmented Lagrangian function of the system (35) where r is a constant [36,31]. Notice that the problem (36) can be generally written into a linear system of equations:…”

Pressure correction methods based on Krylov subspace conception for solving incompressible Navier–Stokes problems

“…In order to improve the convergence behaviour of Uzawa method, Liu and Xu [4] proposed a second-order Richardson algorithm for the saddle point system (8) and (9) by linear combination of Uzawa iterations. It is written as…”

“…It requires almost no additional cost of computation, in terms of storage or CPU time. Numerical tests to the saddle point problems yielded from the Stokes equations which were carried out on the di erent grids showed that this algorithm speeds up convergence of Uzawa algorithm and provides more favourite convergence properties [4].…”