Jacques Rappaz

Finite dimensional approximation of nonlinear problems

SummaryIn the first two papers of this series [4, 5], we have studied a general method of approximation of nonsingular solutions and simple limit points of nonlinear equations in a Banach space. We derive here general approximation results of the branches of solutions in the neighborhood of a simple bifurcation point. The abstract theory is applied to the Galerkin approximation of nonlinear variational problems and to a mixed finite element approximation of the von Kármán equations.

Finite Dimensional Approximation of Non-Linear Problems .1. Branches of Nonsingular Solutions

Note: Univ paris 6,f-75230 paris 05,france. ecole polytech,ctr math appl,f-91128 palaiseau,france. univ pavia,cnr,anal numer lab,i-27100 pavia,italy. Brezzi, f, univ pavia,ist matemat appl,i-27100 pavia,italy.ISI Document Delivery No.: KU350Times Cited: 125Cited Reference Count: 12 Reference ASN-ARTICLE-1980-001doi:10.1007/BF01395985 Record created on 2006-08-24, modified on 2017-05-12SummaryWe begin in this paper the study of a general method of approximation of solutions of nonlinear equations in a Banach space. We prove here an abstract result concerning the approximation of branches of nonsingular solutions. The general theory is then applied to the study of the convergence of two mixed finite element methods for the Navier-Stokes and the von Kármán equations.

Eigenvalue Approximation by Mixed and Hybrid Methods

Note: Univ maryland,dept math,college pk,md 20742. ecole polytech,ctr math appl,f-91128 palaiseau,france. univ paris 6,f-75230 paris 05,france. Mercier, b, cea limeil,f-94190 villeneuve,france.ISI Document Delivery No.: LQ584Times Cited: 29Cited Reference Count: 44 Reference ASN-ARTICLE-1981-005doi:10.2307/2007651 Record created on 2006-08-24, modified on 2017-05-12

Numerical analysis for nonlinear and bifurcation problems

PREFACE Computational applications generally involve nonlinear problems and often contain parameters. They may represent properties of the physical system they describe or quantities which can be varied. A basic problem in approximation consists in studying existence and convergence of approximated solutions for a given nonlinear problem, for instance when the parameters are xed. Another problem is to represent the families or manifolds of solutions under variations of some parameters. Apart from a theoretical approach, such representations are computed and continuation methods are concerned with generating the solution manifolds. By varying one parameter, we can follow a path of solutions. Then to study the eeects of change of parameters on a system, it is of prime interest to know the eeects of numerical approximation on its behavior. The goal of this article is to present a general framework in which approximations of nonlinear problems and approximations of solution manifolds can be studied. We will consider regular solutions, regular solution families, and singular solutions. Even though we will illustrate the general theory only with elementary nite element approximations of model boundary value problems, it can be applied to a much wider range of problems in connection with approximation methods. Our presentation is a remodelling of the one proposed by Crouzeix and Rappaz 1989] taking its origin in Descloux and The general problem we will handle and which covers a lot of applications is the following: nd x 2 X such that F(x) = 0 where X and Z are Banach spaces, F : X ! Z is a smooth nonlinear mapping. Of particular interest is the case where the space X has the form R m Y , where R m with m 1 is the parameter space and the Banach space Y is the state space. We will work under the assumption that the derivative of F is a Fredholm operator of index n 0. Both cases n = 0 and n 1 with a surjective derivative are studied separately. Note that when n is positive, the family of solutions to F(x) = 0 is a diierentiable manifold. The singular situation with a not surjective derivative is also studied. In the general setting, the approximation schemes are written in the form F h (x) = 0 where h is a parameter in (0; 1] and F h : X ! Z is an approximation of F. The family fF h g …

Numerical modeling in induction heating for axisymmetric geometries

This paper deals with numerical simulation of induction heating for axisymmetric geometries. A mathematical model is presented, together with a numerical scheme based on the Finite Element Method. A numerical simulation code was implemented using the model presented in this paper. A comparison between results given by the code and experimental measurements is provided.

Finite Dimensional Approximation of Non-Linear Problems .2. Limit Points

SummaryWe continue here the study of a general method of approximation of nonlinear equations in a Banach space yet considered in [2]. In this paper, we give fairly general approximation results for the solutions in a neighborhood of a simple limit point. We the apply the previous analysis to the study of Galerkin approximations for a class of variationally posed nonlinear problems and to a mixed finite element method for the NavierStokes equations.

NUMERICAL MODELING OF INDUCTION HEATING FOR TWO-DIMENSIONAL GEOMETRIES

We present both a mathematical model and a numerical method for simulating induction heating processes. The geometry of the conductors is cylindrical and the magnetic field is assumed to be parallel to the invariance axis. The model equations have current tension as prescribed data rather than current intensity. In Particular, the formulation of the electromagnetic problem uses the magnetic field as the unknown function. The numerical method takes into account the time periodicity of the prescribed tension and deals with the two different time scales of electromagnetic and thermal phenomena.

Numerical simulation of Rhonegletscher from 1874 to 2100

Due to climatic change, many Alpine glaciers have significantly retreated during the last century. In this study we perform the numerical simulation of the temporal and spatial change of Rhonegletscher, Swiss Alps, from 1874 to 2007, and from 2007 to 2100. Given the shape of the glacier, the velocity of ice u is obtained by solving a 3D nonlinear Stokes problem with a nonlinear sliding law along the bedrock-ice interface. The shape of the glacier is updated by computing the volume fraction of ice @f, which satisfies a transport equation. The accumulation due to snow fall and the ablation due to melting is accounted by adding a source term to the transport equation. A decoupling algorithm allows the two above problems to be solved using different numerical techniques. The nonlinear Stokes problem is solved on a fixed, unstructured finite element mesh consisting of tetrahedrons. The transport equation is solved using a fixed, structured grid of smaller cells. The numerical simulation, from 1874 to 2007, is validated against measurements. Afterwards, three different climatic scenarios are considered in order to predict the shape of Rhonegletscher until 2100. A dramatic retreat of Rhonegletscher during the 21st century is anticipated. Our results contribute to a better understanding of the impact of climatic change on mountain glaciers.

Approximation of the spectrum of a non-compact operator given by the magnetohydrodynamic stability of a plasma

SummaryThe study of the magnetohydrodynamic stability of a plasma leads to a problem of determination of the spectrum of a non-compact selfadjoint operatorA. The spectrum ofA will be approximated by the eigenvalues ofAh, whereAh is a linear operator approximatingA in a finite dimensional space (finite element method) andh is a parameter which tends to zero. Generally the spectrum ofAh “pollutes” spectrum ofA, i.e. for eachh there exists an eigenvalue λh ofAh which, ash tends to zero, converges to λ which is not in the spectrum ofA.We present here a sufficient condition on the finite dimensional spaces used, in order to obtain good approximation properties of the spectrum ofA and, especially, the “non-pollution” property.

Numerical modelling of electromagnetic casting processes

Keywords: BOUNDARY Note: Besson, o, ecole polytech fed lausanne,dept math,ch-1015 lausanne,switzerland.ISI Document Delivery No.: EY619Times Cited: 15Cited Reference Count: 13 Reference ASN-ARTICLE-1991-001doi:10.1016/0021-9991(91)90219-BView record in Web of Science Record created on 2006-08-24, modified on 2017-05-12