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Exact controllability of a pluridimensional coupled problem.

Serge Nicaise — 1992

Revista Matemática de la Universidad Complutense de Madrid

We set a coupled boundary value problem between two domains of different dimension. The first one is the unit cube of Rn, n C [2,3], with a crack and the second one is the crack. this problem comes from Ciarlet et al. (1989), that obtained an analogous coupled problem. We show that the solution has singularities due to the crack. As in Grisvard (1989), we adapt the Hilbert uniqueness method of J.-L. Lions (1968,1988) in order to obtain the exact controllability of the associated wave equation with...

A posteriori error estimates for a nonconforming finite element discretization of the heat equation

Serge NicaiseNadir Soualem — 2005

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

The paper presents an a posteriori error estimator for a (piecewise linear) nonconforming finite element approximation of the heat equation in d , d = 2 or 3, using backward Euler’s scheme. For this discretization, we derive a residual indicator, which use a spatial residual indicator based on the jumps of normal and tangential derivatives of the nonconforming approximation and a time residual indicator based on the jump of broken gradients at each time step. Lower and upper bounds form the main results...

Zienkiewicz–Zhu error estimators on anisotropic tetrahedral and triangular finite element meshes

Gerd KunertSerge Nicaise — 2003

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We consider a posteriori error estimators that can be applied to anisotropic tetrahedral finite element meshes, i.e. meshes where the aspect ratio of the elements can be arbitrarily large. Two kinds of Zienkiewicz–Zhu (ZZ) type error estimators are derived which originate from different backgrounds. In the course of the analysis, the first estimator turns out to be a special case of the second one, and both estimators can be expressed using some recovered gradient. The advantage of keeping two different...

Discrete compactness for a discontinuous Galerkin approximation of Maxwell's system

Emmanuel CreuséSerge Nicaise — 2006

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we prove the discrete compactness property for a discontinuous Galerkin approximation of Maxwell's system on quite general tetrahedral meshes. As a consequence, a discrete Friedrichs inequality is obtained and the convergence of the discrete eigenvalues to the continuous ones is deduced using the theory of collectively compact operators. Some numerical experiments confirm the theoretical predictions.

Boundary stabilization of Maxwell’s equations with space-time variable coefficients

Serge NicaiseCristina Pignotti — 2003

ESAIM: Control, Optimisation and Calculus of Variations

We consider the stabilization of Maxwell’s equations with space-time variable coefficients in a bounded region with a smooth boundary by means of linear or nonlinear Silver–Müller boundary condition. This is based on some stability estimates that are obtained using the “standard” identity with multiplier and appropriate properties of the feedback. We deduce an explicit decay rate of the energy, for instance exponential, polynomial or logarithmic decays are available for appropriate feedbacks.

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