Existence, uniqueness, and approximation of a fictitious domain formulation for fluid-structure interactions

Abstract

In this paper we describe a computational model for the simulation of fluid-structure interaction problems based on a fictitious domain approach. We summarize the results presented over the last years when our research evolved from the Finite Element Immersed Boundary Method (FE-IBM) to the actual Finite Element Distributed Lagrange Multiplier method (FE-DLM). We recall the well-posedness of our formulation at the continuous level in a simplified setting. We describe various time semi-discretizations that provide unconditionally stable schemes. Finally we report the stability analysis for the finite element space discretization where some improvements and generalizations of the previous results are obtained.

On the interface matrix for fluid-structure interaction problems with fictitious domain approach

“…In [9], it was shown that a linearized version of Problem 1 admits a unique solution. See also [8] for a review on the state of the art about this problem.…”

“…In this setting, we are now able to state the finite element version of Problem 3 whose well-posedness has been discussed in [7] and [8].…”

“…As part of our research, several theoretical aspects have been addressed over the years: in [7], we showed the well-posedness of both continuous and discrete problems, in [9] the existence and the uniqueness of the solution were proved in the case of the linearized problem. These results are summarized in a unified setting in [8]. On the other hand, numerical aspects have been studied only from the point of view of time discretization in [11].…”