In this paper we construct a model to describe someaspects of the deformation of the central region of the human lung considered as acontinuous elastically deformable medium. To achieve this purpose, we studythe interaction between the pipes composing the tree and the fluid that goes through it. We use a stationary model to determine the deformed radius of each branch. Then, we solve a constrained minimization problem, so as to minimize the viscous (dissipated) energy in the tree. The key feature of our approach is the useof a fixed point theorem in order to find the optimal flow associatedto a deformed tree. We also give some numerical results withinteresting consequences on human lung deformation during expiration, particularlyconcerning the localization of the equal pressure point (EPP).

We present a fully adaptive multiresolution scheme for spatially one-dimensional quasilinear strongly degenerate parabolic equations with zero-flux and periodic boundary conditions. The numerical scheme is based on a finite volume discretization using the Engquist-Osher numerical flux and explicit time stepping. An adaptive multiresolution scheme based on cell averages is then used to speed up the CPU time and the memory requirements of the underlying finite volume scheme, whose first-order version is known to converge to an entropy solution of the problem. A particular feature of the method is the storage of the multiresolution representation of the solution in a graded tree, whoseleaves are the non-uniform finite volumes on which the numerical divergence is eventually evaluated. Moreover using the L 1 contraction of the discrete time evolution operator we derive the optimal choice of the threshold in the adaptive multiresolution method. Numerical examples illustrate thecomputational efficiency together with the convergence properties.

We propose and analyze numerical schemes for viscosity solutions of time-dependent Hamilton-Jacobi equations on the Heisenberg group.The main idea is to construct a grid compatible with the noncommutative group geometry. Under suitable assumptions on the data, the Hamiltonian and the parameters for the discrete first order scheme,we prove that the error between the viscosity solution computed at the grid nodes and the solution of the discrete problem behaves like $\sqrt{h}$ where h is the mesh step. Such an estimate is similar to those available in the Euclidean geometrical setting.The theoretical results are tested numerically on some examples for which semi-analytical formulas for the computation of geodesics are known. Other simulations are presented, for both steady and unsteady problems.


We prove stability and derive error estimates for the recently introduced central discontinuous Galerkin method, in the context of linear hyperbolic equations with possibly discontinuous solutions. A comparison between the centraldiscontinuous Galerkin method and the regular discontinuousGalerkin method in this context is also made.Numerical experiments are provided to validate the quantitativeconclusions from the analysis.

We propose a numerical scheme to compute the motion of a two-dimensional rigid body in a viscous fluid. Our method combines the method of characteristics with a finite element approximation to solve an ALE formulation of the problem. We derive error estimates implying the convergence of the scheme.

In this work we present new numerical methods to simulate the mechanics of head-tape magnetic storage devices. The elastohydrodynamic problem is formulated in terms of a coupled system which is governed by a nonlinear compressible Reynolds equation for the air pressure over the head, and a rod model for the tape displacement. A fixed point algorithm between the solutions of the elastic and hydrodynamic problems is proposed. For the nonlinear Reynolds equation, a characteristics method and a duality algorithm are developed to cope with the convection dominating and nonlinear diffusion features, respectively. Furthermore, in the duality method the convergence and optimal choice of the parameters are analyzed. At each fixed point iteration, in the elastic model a complementarity formulation is required and appropriate numerical techniques are used. For the spatial discretization different finite element spaces are chosen. Finally, numerical test examples illustrate the theoretical results, as well as the good performance in the simulation of real devices.

In this work, we consider the quasistatic frictionless contact problem between aviscoelastic piezoelectric body and a deformable obstacle. The linear electro-viscoelasticconstitutive law is employed to model the piezoelectric material and the normal compliancecondition is used to model the contact. The variational formulation is derived in a formof a coupled system for the displacement and electric potential fields. An existence anduniqueness result is recalled. Then, a fully discrete scheme is introduced based on thefinite element method to approximate the spatial variable and an Euler scheme to discretizethe time derivatives. Error estimates are derived on the approximative solutions and,as a consequence, the linear convergence of the algorithm is deduced under suitableregularity conditions. Finally, some two-dimensional examples are presented to demonstratethe performance of the algorithm.

We consider the system of partial differential equations governingthe one-dimensional flow of two superposed immiscible layers of shallow water. The difficulty in this system comes from the coupling terms involving some derivatives of the unknowns that make the system nonconservative, and eventually nonhyperbolic. Due to these terms, a numerical scheme obtained by performing an arbitrary scheme to each layer, and using time-splitting or other similar techniques leads to instabilities in general. Here we use entropy inequalities in order to control the stability. We introduce a stable well-balanced time-splitting scheme for the two-layer shallow water system that satisfies a fully discrete entropy inequality. In contrast with Roe type solvers, it does not need the computation of eigenvalues, which is not simple for the two-layer shallow water system. The solver has the property to keep the water heights nonnegative, and to be able to treat vanishing values.