Finite element approximation of multi-scale elliptic problems using patches of elements

In this paper we present a method for the numerical solution of elliptic problems with multi-scale data using multiple levels of not necessarily nested grids. The method consists in calculating successive corrections to the solution in patches whose discretizations are not necessarily conforming. This paper provides proofs of the results published earlier (see C. R. Acad. Sci. Paris, Ser. I 337 (2003) 679–684), gives a generalization of the latter to more than two domains and contains extensive numerical illustrations. New results including the spectral analysis of the iteration operator and a numerical method to evaluate the constant of the strengthened Cauchy-Buniakowski-Schwarz inequality are presented. Supported by CTI Project 6437.1 IWS-IW.     

A least-squares/finite element method for the numerical solution of the Navier–Stokes-Cahn–Hilliard system modeling the motion of the contact line

In this article we discuss the numerical solution of the Navier–Stokes-Cahn–Hilliard system modeling the motion of the contact line separating two immiscible incompressible viscous fluids near a solid wall. The method we employ combines a finite element space approximation with a time discretization by operator-splitting. To solve the Cahn–Hilliard part of the problem, we use a least-squares/conjugate gradient method. We also show that the scheme has the total energy decaying in time property under certain conditions. Our numerical experiments indicate that the method discussed here is accurate, stable and efficient.     

Approximations to solutions of the zakai filtering equation

Mean square estimates are obtained for approximations to solutions of Zakai's equation which depend on the values of the observation process at the points of a regular partition in time     

Systematic study of (Na n )2 dimer transient state in Na n +Na n collisions (n=8, 9, 19, and 20)

From distance dependent tight-binding molecular dynamics simulations, we systematically study the Na _n +Na _n collision dynamics around the first two closed shells (n=8 and 20). We investigate the stability of sodium cluster dimers (Na _n )₂, for many events with random relative orientation at finite temperature, various impact parameters and incident energies. We find that (Na₈)₂, (Na₉)₂, (Na₁₉)₂ and (Na₂₀)₂ can exist during about 3000 fs in central collisions while they can exist up to about ten thousands fs in peripheral collisions with larger impact parameters in fusion mechanism at c.o.m energy per atomE _cm/n=0.025 eV. We observe that the lower the incident energy, the longer the lifetime of the cluster dimers in both central and peripheral collisions. There is no apparent difference in the dynamical stability of (Na₈)₂ and (Na₉)₂, (Na₁₉)₂ and (Na₂₀)₂ although (Na₈)₂ and (Na₂₀)₂ are respectively slightly colder than (Na₉)₂ and (Na₁₉)₂ for the same incident energy per atom and the same impact parameter.     

Three dimensional fluid–structure interaction under pulsatile flow by using distributed Lagrange multiplier method

In this article we discuss an application of a Lagrange multiplier based fictitious domain method for the simulation of moving leaflets in an unsteady flow generated by pressure gradients during the systolic phase of the cardiac cycle in three dimensional geometry. The mathematical model includes the Navier–Stokes equations coupled with the Euler–Newton equations describing fluid–structure interaction for the generalized Neumann boundary conditions on upstream and downstream boundaries. The solution method includes the finite element method combined with an operator-splitting scheme, where the fictitious formulation allows the flow calculations to be in a fixed rectangular parallelepiped to predict the dynamic structural positions and flow field during the valve opening and closing phases.