  The paper discusses a reuse of matrix factorization as a building block in
the Augmented Lagrangian (AL) and modified AL preconditioners for non-symmetric
saddle point linear algebraic systems. The strategy is applied to solve
two-dimensional incompressible fluid problems with efficiency rates independent
of the Reynolds number. The solver is then tested to simulate motion of a
surface fluid, an example of a 2D flow motivated by an interest in lateral
fluidity of inextensible viscous membranes. Numerical examples include the
Kelvin--Helmholtz instability problem posed on the sphere and on the torus.
Some new eigenvalue estimates for the AL preconditioner are derived.
