### Speaker

### Description

Augmented Lagrangian preconditioner for linearized incompressible Navier-Stokes equations has

been introduced in [1,2]. Over the years it was proved to be an efficient technique to solve highly

non-symmetric algebraic systems having a saddle point structure and resulting from discretizations

of fluid problems. In the talk we review the approach and discuss some of its recent developments.

In particular, we introduce the Augmented Lagrangian preconditioner for the algebraic systems of

unfitted finite element discretizations of fluid equations posed on smooth closed surfaces. A matrix

of the system features a sign-indefinite (2,2)-diagonal block with a high-dimensional kernel that

requires special handling. We further consider a reuse of matrix factorization as a building block

in the full and modified Augmented Lagrangian preconditioners. The strategy, applied to solve

two-dimensional incompressible fluid problems, yields efficiency rates independent of the Reynolds

number. The talk partially covers results from [3].

## References

[1] M. Benzi and M. Olshanskii, An augmented Lagrangian approach to linearized problems in hydrodynamic

stability, SIAM J. Sci. Comp. 30 (2005), pp. 1459–1473.

[2] M. Benzi, M. Olshanskii, Z. Wang, Modified augmented Lagrangian preconditioners for the incom-

pressible Navier-Stokes equations, Int. J. Numer. Meth. Fluids 66 (2011), pp. 486–508.

[3] M. Olshanskii and A. Zhiliakov, Recycling augmented Lagrangian preconditioner in an incompressible

fluid solver, Numerical Linear Algebra with Applications 29 (2022), e2415.