Fabien Flori

Fluid-structure interaction : analysis of a 3-D compressible model

Abstract In this paper, we present an existence result of weak solutions for a three-dimensional problem of fluid-plate interaction in which we take into account the non linearity of the continuity equation. This non linearity does not allow, as is usually the case, to neglect the variations of the domain which leads us to study a problem defined on a time dependent domain.

Mathematical and Numerical Analysis of a Tsunami Problem

In this paper, we present a tsunami model based on the displacement of the lithosphere and the mathematical and numerical analysis of this model. More precisely, we give an existence and uniqueness result for a problem which models the flow and formation of waves at the time of a submarine earthquake in the vicinity of the coast. We propose a model which describes the behavior of the fluid using a bi-dimensional shallow-water model by means of a depth-mean velocity formulation. The ocean is coupled to the Earths crust whose movement is assumed to be controlled on a large scale by plate equations. Finally, we give some numerical results showing the formation of a tsunami.

Identification of boundary conditions in a nonlinear shallow water flow

Abstract In this paper, we present sufficient smoothness results on the solution of a nonlinear shallow water problem in order to obtain an existence result for the associated adjoint problem. We introduce a necessary but not sufficient existence condition for the functional minimum. The control is made on the velocity boundary condition, in considering observations at isolated points. Moreover, we give a set of numerical results obtained in a real situation.

Convergence of a characteristic-Galerkin scheme for a shallow water problem

In this paper, we present a numerical method based on a mixed characteristic-Galerkin (or Lagrangian-Galerkin) scheme to solve a shallow water problem with dirichlet boundary conditions. In a first part, we prove an L^2-bound on the water elevation. This bound is obtained by Lions in the case of a linearized momentum equation ^[^1^] and we extend it to the nonlinear case for which the existence is shown for small data ^[^2^]. This bound allows us to construct solutions as limits of the solutions of a regularized problem and to prove the convergence of the discrete problem towards the continuous. We give a numerical criteria connecting the Lagragian discretization and the number of Galerkin eigenvectors to solve the discrete equations with a fixed-point procedure. We present a few numerical results in the case of a fixed domain showing the coherence of the scheme which seems to be adaptable to a domain depending on time.

A numerical method for a blood-artery interaction problem

We present a numerical method for the resolution of a bidimensional blood flow problem and more generally for a fluid flow surrounded by a time dependent domain. Our approach is based on an ALE formulation which is solved using a Galerkin method with an eigenvectors basis set on the initial fixed domain.