This paper deals with the numerical approximation of mild solutions of elliptic-parabolic equations, relying on the existence results of Bénilan and Wittbold (1996). We introduce a new and simple algorithm based on Halpern's iteration for nonexpansive operators (Bauschke, 1996; Halpern, 1967; Lions, 1977), which is shown to be convergent in the degenerate case, and compare it with existing schemes (Jäger and Kačur, 1995; Kačur, 1999).

We analyze the compressible isentropic Navier–Stokes equations (Lions, 1998) in the two-dimensional case with $\gamma=\displaystyle{{c_{p}}/{c_{v}}}=2$ . These equations also modelizethe shallow water problem in height-flow rate formulation used tosolve the flow in lakes and perfectly well-mixed sea. We establisha convergence result for the time-discretized problem when themomentum equation and the continuity equation are solved with theGalerkin method, without adding a penalization term in thecontinuity equation as it is made in Lions (1998). The secondpart is devoted to the numerical analysis and mainly deals withproblems of geophysical fluids. We compare the simulationsobtained with this compressible isentropic Navier–Stokes model andthose obtained with a shallow water model (Di Martino et al., 1999). At first,the computations are executed on a simplified domain in order tovalidate the method by comparison with existing numerical resultsand then on a real domain: the dam of Calacuccia (France). At last, we numerically implement an analyticalexample presented by Weigant (1995) which shows thateven if the data are rather smooth, we cannot have bounds onρ in Lp for p large if $\gamma<2$ when N=2.

We present here a discretization of a nonlinear obliquederivative boundary value problem for the heat equation in dimensiontwo. This finite difference scheme takes advantages of thestructure of the boundary condition, which can be reinterpreted as aBurgers equation in the space variables. This enables to obtain anenergy estimate and to prove the convergence of the scheme. We also provide some numerical simulations of thisproblem and a numerical study of the stability of the scheme, whichappears to be in good agreement with the theory.

A numerically inexpensive globalization strategy of sequential quadratic programming methods (SQP-methods) for control of the instationary Navier Stokes equations is investigated. Based on the proper functional analytic setting a convergence analysis for the globalized methodis given. It is argued that the a priori formidable SQP-step can be decomposed into linear primal and linear adjoint systems, which is amenable for existing CFL-software. A report on a numerical test demonstrates the feasibility of the approach.

During the development of a parallel solver for Maxwell equations by integral formulations and Fast Multipole Method (FMM), we needed to optimize a critical part including a lot of communications and computations. Generally, many parallel programs need to communicate, but choosing explicitly the way and the instant may decrease the efficiency of the overall program. So, the overlapping of computations and communications may be a way to reduce this drawback. We will see a implementation of this techniques using dynamic and adaptive overlapping based on the EasyMSG high level C++ library over MPI, a case of SPMD programming.

We consider a new formulation for finite volume element methods, which is satisfied byknown finite volume methods and itcan be used to introduce new ones. This framework results by approximating the test function in theformulation of finite element method.We analyze piecewise linear conforming or nonconforming approximations on nonuniform triangulations andprove optimal order H 1-norm and L 2-norm errorestimates.

We describe both the classical Lagrangian and the Eulerian methods for first order Hamilton–Jacobi equations of geometric optic type. We then explain the basic structure of the softwareand how new solvers/models can be added to it. A selection of numerical examples are presented.

The present paper is devoted to the computation of single phase or two phase flows using the single-fluid approach. Governing equations rely on Euler equations which may be supplemented by conservation laws for mass species. Emphasis is given on numerical modelling with help of Godunov scheme or an approximate form of Godunov scheme called VFRoe-ncv based on velocity and pressure variables. Three distinct classes of closure laws to express the internal energy in terms of pressure, density and additional variables are exhibited. It is shown first that a standard conservative formulation of above mentioned schemes enables to predict “perfectly” unsteady contact discontinuities on coarse meshes, when the equation of state (EOS) belongs to the first class. On the basis of previous work issuing from literature, an almost conservative though modified version of the scheme is proposed to deal with EOS in the second or third class. Numerical evidence shows that the accuracy of approximations of discontinuous solutions of standard Riemann problems is strengthened on coarse meshes, but that convergence towards the right shock solution may be lost in some cases involving complex EOS in the third class. Hence, a blend scheme is eventually proposed to benefit from both properties (“perfect” representation of contact discontinuities on coarse meshes, and correct convergence on finer meshes). Computational results based on an approximate Godunov scheme are provided and discussed.

The aim of this work is to deduce the existence of solutionof a coupled problem arising in elastohydrodynamiclubrication. The lubricant pressure and concentration aremodelled by Reynolds equation, jointly with the free-boundaryElrod-Adams model in order to take into account cavitationphenomena. The bearing deformation is solution of Koitermodel for thin shells. The existence of solution to thevariational problem presents some difficulties: the coupledcharacter of the equations, the nonlinear multivaluedoperator associated to cavitation and the fact of writing theelastic and hydrodynamic equations on two different domains.In a first step, we regularize the Heaviside operator.Additional difficulty related to the differentdomains is circumvented by means of prolongation andrestriction operators, arriving to a regularized coupledproblem. This one is decoupled into elastic and hydrodynamicparts, and we prove the existence of a fixed point for theglobal operator. Estimations obtained for theregularized problem allow us to prove the existence ofsolution to the original one. Finally, a numerical method is proposed in orderto simulate a real journal-bearing device and illustrate the qualitative andquantitative properties of the solution.

This paper is devoted to the study of a turbulentcirculation model. Equations are derived from the “Navier-Stokes turbulentkinetic energy” system. Some simplifications are performed but attentionis focused on non linearities linked to turbulent eddy viscosity  $\nu _{t}$ . The mixing length $\ell $ acts as a parameter which controls theturbulent part in $\nu _{t}$ . The main theoretical results that we haveobtained concern the uniqueness of the solution for bounded eddy viscositiesand small values of $\ell $ and its asymptotic decreasing as $\ell\rightarrow \infty $ in more general cases. Numerical experimentsillustrate but also allow to extend these theoretical results: uniqueness isproved only for $\ell $ small enough while regular solutions are numericallyobtained for any values of $\ell $ . A convergence theorem is proved forturbulent kinetic energy: $k_{\ell }\rightarrow 0$ as $\ell \rightarrow\infty ,$ but for velocity $u_{\ell }$ we obtain only weaker results.Numerical results allow to conjecture that $k_{\ell }\rightarrow 0,$ $\nu_{t}\rightarrow \infty $ and $u_{\ell }\rightarrow 0$ as $\ell \rightarrow\infty .$ So we can conjecture that this classical turbulent model obtainedwith one degree of closure regularizes the solution.

The Boltzmann–Poisson system modeling the electron flow in semiconductorsis used to discuss the validity of the Child–Langmuir asymptotics.The scattering kernel is approximated by a simple relaxation time operator.The Child–Langmuir limit gives an approximation of the current-voltagecharacteristic curves by means of a scalingprocedure in which the ballistic velocity is much larger that the thermal one. We discuss the validity of the Child–Langmuir regime by performing detailed numerical comparisons between the simulation of theBoltzmann–Poisson system and the Child–Langmuir equations in testproblems.

This article presents the guiding principles of the architecture of Trio_U, a new generation of software for thermohydraulic calculations. Trio_U is designed to serve as a thermohydraulic development platform. Its basic conception is object-oriented and it is written in C++. The article demonstrates how this type of design enables an open, modular software architecture.

A finite element code, called ZéBuLoN was parallelised some years ago. This code is entirely written using an object oriented framework (C++ is the support language). The aim of this paper is to present some problems which arose during the parallelization, and some innovative solutions. Especially, a new concept of message passing is presented which allows to take into account SMP machines while still using the parallel virtual machine abstraction.

Along with the classical requirements on B-splines bases(minimal support, positivity, normalization)we show that it is natural to introduce an additional“end point property". When dealing with multiple knots,this additional property is exactly the appropriate requirementto obtain the poles of nondegenerate splinesas intersections of osculating flats at consecutive knots.

Object oriented design has proven itself as a powerful tool in the field of scientific computing. Several software packages, libraries and toolkits exist, in particular in the FEM arenathat follow this design methodology providing extensible, reusable,and flexible software while staying competitive to traditionallydesigned point tools in terms of efficiency. However, the common approach to identify classes is to turn data structures and algorithms of traditional implementations into classes such that the level of abstraction is essentially not raised. In this paper we discuss an alternative way to approach the design challenge which we call “concept oriented design”. We apply this design methodology to Petrov-Galerkin methods leading to a class library for both, boundary element methods (BEM) and finite element methods (FEM). We show as a particular example the implementation of hp-FEM using the library with specialattention to the handling of inconsistent meshes.

In this paper we test the feasibility of coupling two heterogeneous mathematical modeling integrated within two different codes residing on distant sites. A prototype is developed using Schwarz type domain decomposition as the mathematical tool for coupling. The computing technology for coupling uses a CORBA environment to implement a distributed client-server programming model. Domain decomposition methods are well suited to reducing complex physical phenomena into a sequence of parallel subproblems in time and space. The whole process is easily tuned to underlying hardware requirements.