Publications by Jacques Rappaz

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The bending terms in a shell are small with respect to membrane ones as the thickness tends to zero. Consequently, the membrane approximation gives a good description of vibration properties of a thin shell. This vibration problem is associated with a non-compact resolvent operator, and spectral pollution could appear when computing Galerkin approximations. That is to say, there could exist sequences of eigenvalues of the approximate problems that converge to points of the resolvent set of the exact problem

In order to numerically solve the interior and the exterior Dirichlet problems for the Laplacian operator, we have presented in a previous paper a method which consists in inverting, on a finite element space, a non-singular integral operator for circular domains. This operator was described as a geometrical perturbation of the Steklov operator, and we have precisely defined the relation between the geometrical perturbation and the dimension of the finite element space, in order to obtain a stable and conve

In this paper, a nonlinear problem corresponding to a simplified Oldroyd-B model without convective terms is considered. Assuming the domain to be a convex polygon, existence of a solution is proved for small relaxation times. Continuous piecewise linear finite elements together with a Galerkin Least Square (GLS) method are studied for solving this problem. Existence and a priori error estimates are established using a Newton-chord fixed point theorem, a posteriori error estimates are also derived. An Elast

We develop a discretization and solution technique for elliptic problems whose solutions may present strong variations, singularities, boundary layers and oscillations in localized regions. We start with a coarse finite element discretization with a mesh s ...

In this paper we study existence and uniqueness of weak solutions for some non-linear weighted Stokes problems using convex analysis. The characteri- zation of these considered equations is that the viscosity depends on the strain rate of the velocity field with a weight being a positive power of the distance to the boundary of the domain. These non-linear relations can be seen as a first approach of mixing-length eddy viscosity from turbulent modeling. A well known model is von Karman’s on which the viscos

The aim of this work is to derive rate of convergence estimates for the spectral approximation of a mathematical model which describes the vibrations of a solid-fluid type structure. First, we summarize the main theoretical results and the discretization of this variational eigenvalue problem. Then, we state some well known abstract theorems on spectral approximation and apply them to our specific problem, which allow us to obtain the desired spectral convergence. By using classical regularity results, we a

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. ...

The gas bubble laden layer under the anodes during the electrolysis of alumina plays an important role in the hydrodynamics and the voltage balance of the reduction cells. Under certain geometrical and operational conditions, very large gas pockets, in the order of hundred cubic centimeters can be formed. The particular shape of these large gas bubbles was first described by Fortin et al in 1984 [1]. In the present paper the results of a combined experimental and numerical approach are described. In the exp

A priori and a posteriori error estimates are presented for nonlinear diffusion-convection problems when using the classical Streamline Upwind Petrov-Galerkin (SUPG) scheme and numerical integration. For this purpose, an abstract framework is developed. A priori estimates are derived in the H-1 and L(2) norms and the error is bounded above and below by an estimator based on the equation residual. An adaptive algorithm requiring the generation of successive Delaunay triangulations is proposed and numerical r

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.

The motion of a two-dimensional glacier is considered. At each time step, given the shape of the glacier, ice is modelled as an incompressible non-Newtonian fluid and a non-linear elliptic problem has to be solved to obtain the horizontal velocity field. Then, the upper surface of the glacier is updated by solving a transport equation. Finite element techniques are used to compute the velocity field whereas the transport equation is solved using a Lax-Wendroff scheme. Numerical results are compared to exper

In this paper we present a family of iterative methods to solve numerically second order elliptic problems with multi-scale data using multiple levels of grids. These methods are based upon the introduction of a Lagrange multiplier to enforce the continuity of the solution and its fluxes across interfaces. This family of methods can be interpreted as a mortar element method with complete overlapping domain decomposition for solving numerically multi-scale elliptic problems. (C) 2004 Academie des sciences. P

Our goal is to study the existence of weak solution for the stationary system of equations describing the solidification process of a binary alloy confined in a bounded regular domain in R-3 and having mixed boundary conditions. (c) 2005 Elsevier Ltd. All ...

In an abstract framework we present a formalism which specifies the notions of consistency and stability of Petrov-Galerkin methods used to approximate nonlinear problems which are, in many practical situations, strongly nonlinear elliptic problems. This formalism gives rise to a priori and a posteriori error estimates which can be used for the refinement of the mesh in adaptive finite element methods applied to elliptic nonlinear problems. This theory is illustrated with the example: -div (k(u)DELTAu) + c.

We consider a two-dimensional Magnetohydrodynamic system of equations describing the motion of a conducting fluid in which eddy currents flow. The mathematical model is derived and existence of solutions is proved by using fixed point techniques. Uniqueness is obtained under restrictive conditions on the involved physical parameters.

In this paper we present a method to solve numerically 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. It resembles the FAC method (see Math. Comp. 46 (174) (1986) 439-456) and its convergence is obtained by a domain decomposition technique (see Math. Comp. 57 (195) (1991) 1-21). However it is of much more flexible use in co

In this article, we present a numerical scheme based on a finite element method in order to solve a time-dependent convection-diffusion equation problem and satisfy some conservation properties. In particular, our scheme is able to conserve the total energ ...

Numerical simulation in aluminum production envolves many different topics, this paper is dedicated to some aspects of two of them. In the first part, a numerical approach for the computation of the motion of the fluids in an electrolitic cell is presented ...

We present an extension of a 2D cartesian electromagnetic and thermal model to achieve the optimal control of an industrial induction heating system. The control variables are the frequency and the power. We explain how to derive the discrete optimal control algorithm and show the numerical results we obtained for a specific case.

The main goal of this article is to analyze a three-dimensional model for stress and velocity fields in grounded glaciers and ice sheets including the role of normal deviatoric stress gradients. This model leads to a nonlinear system of stationary partial ...

Jacques Rappaz

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On spectral pollution in the finite element approximation of thin elastic ''membrane'' shells

Numerical analysis of a non-singular boundary integral method: Part II: The general case

Existence, a priori and a posteriori error estimates for a nonlinear three-field problem arising from Oldroyd-B viscoelastic flows

Multiscale algorithm with patches of finite elements

On Some Weighted Stokes Problems. Application on Smagorinsky Models

Rate of convergence estimates for the spectral approximation of a generalized eigenvalue problem

Numerical simulation of Rhonegletscher from 1874 to 2100

Large Gas Bubbles under the Anodes of Aluminum Electrolysis Cells

Error estimates and adaptive finite elements for nonlinear diffusion-convection problems

Numerical modeling in induction heating for axisymmetric geometries

Numerical simulation of the motion of a two-dimensional glacier

A multi-domain method for solving numerically multi-scale elliptic problems

Weak solution of a stationary convection-diffusion model describing binary alloy solidification processes

Consistency, Stability, a-Priori and a-Posteriori Errors for Petrov-Galerkin Methods Applied to Nonlinear Problems

On a 2-Dimensional Magnetohydrodynamic Problem .1. Modeling and Analysis

Approximation of multi-scale elliptic problems using patches of finite elements

Conservation Schemes For Convection-Diffusion Equations With Robin Boundary Conditions

Scientific Computing For Aluminum Production

Power formulation for the optimal control of an industrial induction heating process for thixoforming

Mathematical and numerical analysis of a three-dimensional fluid flow model in glaciology