Jean Descloux

On Finite Element Matrices

This paper considers matrices arising from the use of finite element techniques in least-squares approximation and in elliptic partial differential equations; it studies their properties of numerical stability, and in particular, it establishes bounds for their inverses with respect to the uniform norm.

An Accurate Algorithm for Computing the Eigenvalues of a Polygonal Membrane

For a polygonal domain Ω, we consider the eigenvalue problem Δu + λu = 0 in Ω, u = 0 on the boundary of Ω. Ω is decomposed into subdomains Ω1, Ω2,...; on each Ωi, u is approximated by a linear combination of functions which satisfy the equation Δu + Δu = 0 and continuity conditions are imposed at the boundaries of the subdomains. We propose a non-conventional method based on the use of a Rayleigh quotient. We present numerical examples and a proof of the exponential convergence of the algorithm.

A Stability Result for the Magnetic Shaping Problem

We consider the bidimensional magnetic shaping problem without surface tension and study its stability when the boundary Γ of the domain has cusp points. We show in particular that one has stability when the curvature of the smooth parts of Γ is negative.

Essential Numerical Range of an Operator with Respect to a Coercive Form and the Approximation of Its Spectrum by the Galerkin Method

We define the notion of essential numerical range of an operator with respect to a coercive sesquilinear form and we prove convergence of the Galerkin approximate eigenvalues which lie outside this region.

Approximation of the Spectrum of Closed Operators - the Determination of Normal-Modes of a Rotating Basin

Note: Univ michigan,dept math,ann arbor,mi 48109. ecole polytech,ctr math appl,f-91128 palaiseau,france. Descloux, j, ecole prat hautes etud,dept math,f-75231 paris 05,france.ISI Document Delivery No.: LE068Times Cited: 3Cited Reference Count: 17 Reference ASN-ARTICLE-1981-004doi:10.2307/2007731 Record created on 2006-08-24, modified on 2017-05-12

A two fluids stationary free boundary problem

Abstract We consider the stationary flow in a cavity of two immiscible incompressible Navier-Stokes fluids. The superficial tension forces are neglected. We propose a numerical algorithm for solving this free boundary problem.

A Nonlinear Inverse Power Method with Shift

The classical inverse power method for linear eigenvalue problems is generalized for some nonlinear equations. The algorithm does not involve the evaluation of Jacobian operators.

On Hopf and Subharmonic Bifurcations

Let (λ,x) ∈ ℝ×ℝn →f(λ,x) ∈ ℝn be a given function that, for simplicity, we shall assume to be of classe C∞. We also assume that the partial derivative Dxf(λ,x) is bounded, uniformly with respect to (λ,x) ∈ ℝ × ℝn.

On the heat equation

Our aim is to construct a mathematical model that describes temperature distribution in a body via heat conduction. There are other forms of heat transfer such as convection and radiation that will not be considered here. Basically, heat conduction in a body is the exchange of heat from regions of higher temperatures into regions with lower temperatures. This exchange is done by a transfer of kinetic energy through molecular or atomic vibrations. The transfer does not occur at the same rate for all materials. The rate of transfer is high for some materials and low for others. This thermal diffusivity depends mainly on the atomic structure of the material. To interpret this mathematically, we first need to recall the notion of flux (that you might have seen in multivariable calculus). Suppose that a certain physical quantity Q flows in a certain region of the 3-dimensional space R. For example Q could represent a mass (think of flowing water), or could represent energy (think of heat), or an electric charge. The flux corresponding to Q is a vector-valued function q⃗ whose direction indicates the direction of flow of Q and whose magnitude |q⃗| the rate of change of Q per unit of area per unit of time. If for example Q measures gallons of water, then the units for the flux could be gallons per meter per minute. One way to understand the relationship between Q and q⃗ is a as follows. Let m0 = (x0, y0, z0) be a point in R with the standard canonical basis of orthonormal vectors i⃗, j⃗, k⃗. Consider a small rectangular surface S1 centered at m0 and parallel to the yz-plane. So the unit vector i⃗ is normal to S1. Assume that S1 has side lengths ∆z and ∆y (see Figure 1.)

On a Finite-Element Method to Solve the Criticality Eigenvalue Problem for the Transport-Equation

A finite element discretization of the criticality eigenvalue problem for a one-dimensional model of the transport equation is analyzed. The existence of spurious eigenvalues for this procedure is demonstrated. However, it is shown that all of the spurious eigenvalues are larger than the smallest correct eigenvalue. This gives a justification for the use of the inverse power method to solve the discretized problem for the criticality eigenvalue, which is the smallest eigenvalue.