Nguyenho Ho
Accelerated Uzawa Iteration for the Stokes Equations

Mathematical Sciences Department
Worcester Polytechnic Institute
100 Institute Road
Worcester
MA 01609-2280
nho@wpi.edu
Sarah Olsen
Homer Walker

The finite-element discretization of the Stokes equations leads to a saddle-point problem

$$\left[\begin{array}{cc} {A} & {B}^T \ {B}& 0 \end{array} \right]... ... p \end{array} \right]= \left[\begin{array}{c} {\bf f} \ 0\end{array} \right]$$ (1)

where $A$ is symmetric positive-definite and $B$ is full-rank. Saddle-point systems of this type arise from many sources, and their numerical solution has been extensively studied (see, e.g., [2]). Here, we focus on the Stokes system (1) obtained using a P2/P0 finite-element discretization and consider the Uzawa iteration [3]

$$\begin{array}{rcl} A {\bf u}_{k+1} &=& {\bf f} - B^T p_k \ p_{k+1} &=& p_k + \omega B {\bf u}_{k+1}. \end{array}$$

Regarding this as a fixed-point iteration on ${\bf u}$ and $p$ , we augment it with Anderson acceleration [1] to improve the convergence. We show the results of a numerical study in which we compare the performance in several test cases of Uzawa iteration with and without Anderson acceleration as well as several alternative solution approaches.

20pt[1] D. G. Anderson, Iterative procedures for nonlinear integral equations, J. Assoc. Comput. Mach., 12 (1965), pp. 547-560.

20pt[2] M. Benzi, G. H. Golub, and J. Liesen, Numerical solution of saddle point problems, Acta Numerica (2005), Cambridge University Press, 2005, pp. 1-137,

20pt[3] H. Uzawa, Iterative methods for concave programming, in Studies in Linear and Nonlinear Programming, K. J. Arrow, L. Hurwicz, and H. Uzawa, eds., Stanford University Press, Stanford, CA, 1958, pp. 154-165.