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

We are interested in solving the system (1)[ALTL0][cλ]=[FG], 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 ∈ Rn, L is a (m,n) matrix, and λ,G∈Rm. 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.