On convergence rate of the augmented Lagrangian algorithm for nonsymmetric saddle point problems

We are interested in solving the system [ A LT L 0 ][ c λ ] = [ F G ] , (1) by a variant of the augmented Lagrangian algorithm. This type of problem with nonsymmetric A typically arises in certain discretizations of the Navier–Stokes equations. Here A is a (n,n) matrix, c, F ∈ R, L is a (m,n) matrix, and λ,G ∈ R. We assume that A is invertible on the kernel of L. Convergence rates of the augmented Lagrangian algorithm are known in the symmetric case but the proofs in [R. Glowinski, P. LeTallec, Augmented Lagrangian and Operator Splitting Methods in Nonlinear Mechanics, SIAM, 1989] used spectral arguments and cannot be extended to the nonsymmetric case. The purpose of this paper is to give a rate of convergence of a variant of the algorithm in the nonsymmetric case. We illustrate the performance of this algorithm with numerical simulations of the lid-driven cavity flow problem for the 2D Navier–Stokes equations.  2004 IMACS. Published by Elsevier B.V. All rights reserved.