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The corrected Uzawa method for solving saddle point problems

In this paper, we consider the solution of linear systems of saddle point type by correcting the Uzawa algorithm, which has been proposed in [K. Arrow, L. Hurwicz, H. Uzawa, Studies in nonlinear programming, Stanford University Press, Stanford, CA, 1958]. We call this method as corrected Uzawa (CU) method. The convergence of the CU method is analyzed for solving nonsingular saddle point problem as well as the semi‐convergence for the singular case. First, the corrected model for the Uzawa algorithm is established, and the CU algorithm is presented. Then we study the geometric meaning of the CU model. Moreover, we introduce the overall reduction coefficient α to measure the effect of the CU process. It is shown that the CU method converges faster than the Uzawa method and several other methods if the overall reduction coefficient α satisfies certain conditions. Numerical experiments are presented to illustrate the theoretical results and examine the numerical effectiveness of the CU method. Copyright © 2015 John Wiley & Sons, Ltd.