Jean-Claude Nédélec

Mixed finite elements in ℝ 3

SummaryWe present here some new families of non conforming finite elements in ℝ3. These two families of finite elements, built on tetrahedrons or on cubes are respectively conforming in the spacesH(curl) andH(div). We give some applications of these elements for the approximation of Maxwells equations and equations of elasticity.

A Preconditioner for the Electric Field Integral Equation Based on Calderon Formulas

We describe a preconditioning technique for the Galerkin approximation of the electric field integral equation (EFIE), which arises in the scattering theory for harmonic electromagnetic waves. It is based on a discretization of the Calderon formulas and the Helmholtz decomposition. We prove several properties of the method, in particular that it produces a variational solution on a subspace of the Galerkin space for which we have an LBB inf-sup condition. When the Krylov spaces associated with the continuous operators are nondegenerate we prove that the discrete Krylov spaces converge as the mesh refinement goes to zero; when, moreover, the EFIE is nondegenerate on the continuous Krylov spaces, the discrete Krylov iterates converge towards the continuous ones. We also argue that one might expect the continuous Krylov iterates to exhibit superlinear convergence of the form

Éléments finis mixtes incompressibles pour l'équation de Stokes dans ℝ3

n \mapsto C^n(n!)^{-\alpha}

Electromagnetic waves in an inhomogeneous medium

for some C > 0 and

Low-frequency electromagnetic scattering

\alpha>0

Regularization in 3D for Anisotropic Elastodynamic Crack and Obstacle Problems

. Finally, we illustrate the theory with numerical experiments.

Computing numerically the Green’s function of the half-plane Helmholtz operator with impedance boundary conditions

SummaryWe introduce some new families of finite element approximation for the stationary Stokes and Navier Stokes equations in a bounded domain in ℝ3. These elements can used tetahedrons or cubes. The approximation satisfie exactly the incompressibility condition.

Numerical stability in the calculation of eigenfrequencies using integral equations

Abstract In this paper we consider the electromagnetic wave problem in an inhomogeneous medium. We first prove uniqueness of the solution using Rellich and Cauchy-Kowalewska theorems. Then we explicitly compute the Dirichlet-Neumann operator on the sphere, we reduce the equations to a problem on a truncated domain, and we give a variational formulation. This formulation reads as a compact perturbation of a coercive operator, which leads to the existence of the solution according to Fredholms alternative.

An Efficient Galerkin BEM to Compute High Acoustic Eigenfrequencies

The main result of this paper is to reduce the calculation of higher-order terms in the asymptotic expansions of the electric and magnetic fields at low frequencies to the solutions of certain canonical problems. Our approach is based on coupling the power series representation of the scattered fields with expansion of the exact nonlocal radiation condition. We also provide a new and simple variational proof of the convergence of the electric and magnetic fields solutions of the scattering problem for the Maxwell equations as the frequency goes to zero. Besides its theoretical interest, our analysis is motivated by its application to the numerical computation of the higher-order terms. These higher-order terms may be combined to Pade approximations to enlarge the domain of applicability of the low-frequency scattering to predict more accurately the reponse of diffraction problems for heteregeneous Maxwells equations in the resonance region where the wavelength and the dimension of the dielectric material...

Approximation numérique du cisaillement transverse dans les plaques minces en flexion

The problems of wave scattering by obstacles or cracks appear very often in geophysics and in mechanics. In particular the linearized theory of elastodynamics for 3 dimensional elastic material is used frequently, because this theory keeps the analysis relatively simple. Even with this theory, however, a practical analysis is possible only with the use of some numerical methods. This has been the raison d’etre of many numerical experiments carried out in the engineering community. Among those numerical methods tested so far, the boundary integral equation (BIE) method has been accepted favourably by engineers, presumably because it can deal with scattered waves effectively in external problems. In particular the double layer potential representation is considered to be an efficient tool of numerical analysis for wave problems including cracks. The only inconvenience of the double layer potential approach, however, is the hypersingularity of the kernel, which does not permit the use of conventional numerical integration techniques. Hence we can take advantage of this approach only after weakening the hypersingularity of the kernel, or only after ‘regularizing’ it. As a matter of fact, some of such attemps can be found in the articles by Sladek & Sladek [11], Bui [5], Bonnet [4], Polch et.al [10], Nishimura & Kobayashi [8], [9] who used the collocation method and in Nedelec [7], Bamberger [1] where the variational method has been used.