We present some recent results on the finite element approximation of fluid-structure interaction problems. In particular, we consider a new variational formulation of the Immersed Boundary Method (IBM) based on the introduction of a suitable Lagrange multiplier. We prove that a semi-implicit time advancing scheme is unconditionally stable. At each time step we have to solve a saddle point problem. We show existence and uniqueness of the continuous solution and analyze the finite element discretization providing optimal error estimates.

For more details see:
- D. Boffi, N. Cavallini, L. Gastaldi: The Finite Element Immersed Boundary Method with Distributed Lagrange multiplier, SIAM J. of Numerical Analysis 53 (2015) pp. 2584-2604
- D. Boffi, L. Gastaldi: A fictitious domain approach with Lagrange multiplier for fluid-structure interactions, Numer. Math. 135 (2017) pp.711-732 
- D. Boffi, L. Gastaldi, L. Heltai: A distributed Lagrange formulation of the Finite Element Immersed Boundary Method for fluids interacting with compressible solids, arXiv:1712.02545 [math.NA]

Lucia Gastaldi is full professor of Numerical Analysis at University of Brescia, Department of Civil, Environmental, Architectural Engineering and Mathematics since 1997.
Scientific Interest: finite element approximation of partial differential equations; fluidstructure interactions; immersed boundary method; mixed finite elements; approximation of eigenproblems in mixed form; adaptive finite element method.
Author of more than 60 papers published in international journals. Prof. Gastaldi presented a communication within several national and international conferences, some of them invited. She organized sessions during international conferences and in 2006 the CIME summer course "Mixed Finite Elements, Compatibility Conditions, and Applications".
Prof. Gastaldi was visiting professor at Centro de Matematica e Applicacoes Fundamentais, Lisbona, "IMA Institute for Mathematics and Its Applications", Minneapolis, USA, and "Department of Mathematics", University of Buenos Aires, Argentina.
She is a member of the Editorial Board of CAMWA and CAS and referee for international journals.