Jun 10 – 11, 2022
Dipartimento di Matematica, Università di Pisa
Europe/Rome timezone

Augmented Lagrangian preconditioner for fluids on surfaces

Jun 11, 2022, 11:15 AM
30m
Aula Magna (Dipartimento di Matematica, Università di Pisa)

Aula Magna

Dipartimento di Matematica, Università di Pisa

L.go B. Pontecorvo, 5 56127 Pisa (PI) Italy

Speaker

Maxim Olshanskii (Department of Mathematics, University of Houston)

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.

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