Optimal and Scalable Augmented Lagrangian preconditioners for Fictitious Domain problems

by Federica Mugnaioni (Scuola Normale Superiore)

Tuesday 13 May 2025, 11:00 → 12:00 Europe/Rome

Saletta Riunioni (Dipartimento di Matematica)

One of the major drawbacks of using Fictitious Domain methods is the computational demands of solving the associated large-scale linear systems, both in terms of time and memory. To address this issue, we propose two augmented Lagrangian-based preconditioners for efficiently solving linear systems of equations with a block two-by-two and three-by-three structure arising from fictitious domain problems and from finite element discretizations of immersed boundary methods. We consider two relevant examples to illustrate the performance of these preconditioners when used in conjunction with flexible GMRES: the Poisson and the Stokes fictitious domain problems.  We provide a detailed spectral analysis, deriving lower and upper bounds for the eigenvalues of the preconditioned matrix and showing their independence with respect to discretization parameters. Furthermore, we discuss the eigenvalue distribution when inexact versions of the preconditioners are employed. We show the effectiveness of the proposed approach and the robustness of our preconditioning strategies through extensive numerical tests in both two and three dimensions, using different immersed geometries.

• M. Benzi, M. Feder, L. Heltai and F. Mugnaioni. Optimal and Scalable Augmented Lagrangian preconditioners for Fictitious Domain problems. arXiv:2504.11339, 2025