Well-posedness and Robust Preconditioners for the Discretized Fluid-Structure Interaction Systems

In this paper we develop a family of preconditioners for the linear algebraic systems arising from the arbitrary Lagrangian-Eulerian discretization of some fluid-structure interaction models. After the time discretization, we formulate the fluid-structure interaction equations as saddle point problems and prove the uniform well-posedness. Then we discretize the space dimension by finite element methods and prove their uniform well-posedness by two different approaches under appropriate assumptions. The uniform well-posedness makes it possible to design robust preconditioners for the discretized fluid-structure interaction systems. Numerical examples are presented to show the robustness and efficiency of these preconditioners.Comment: 1. Added two preconditioners into the analysis and implementation 2. Rerun all the numerical tests 3. changed title, abstract and corrected lots of typos and inconsistencies 4. added reference

FIELD-OF-VALUES CONVERGENCE ANALYSIS OF AUGMENTED LAGRANGIAN PRECONDITIONERS FOR THE LINEARIZED NAVIER-STOKES PROBLEM

We study a block triangular preconditioner for finite element approximations of the linearized Navier–Stokes equations. The preconditioner is based on the augmented Lagrangian formulation of the problem and was introduced by the authors in [2]. In this paper we prove field-ofvalues type estimates for the preconditioned system which lead to optimal convergence bounds for the GMRES algorithm applied to solve the system. Two variants of the preconditioner are considered: an ideal one based on exact solves for the velocity submatrix, and a more practical one based on block triangular approximations of the velocity submatrix