About the Lamé system in a polygonal or a polyhedral domain and a coupled problem between the Lamé system and the plate equation. II : exact controllability
About the Lamé system in a polygonal or a polyhedral domain and a coupled problem between the Lamé system and the plate equation. I : regularity of the solutions
Exact controllability of a pluridimensional coupled problem.
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
The paper presents an a posteriori error estimator for a (piecewise linear) nonconforming finite element approximation of the heat equation in , 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
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...
Control of networks of Euler-Bernoulli beams
Boundary stabilization of Maxwell’s equations with space-time variable coefficients
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.
Coefficients of the singularities on domains with conical points
As a model for elliptic boundary value problems, we consider the Dirichlet problem for an elliptic operator. Solutions have singular expansions near the conical points of the domain. We give formulas for the coefficients in these expansions.
Singularities of Maxwell’s system in non-hilbertian Sobolev spaces
We study the regularity of the solution of the regularized electric Maxwell problem in a polygonal domain with data in . Using a duality method, we prove a decomposition of the solution into a regular part in the non-Hilbertian Sobolev space and an explicit singular one.
Energy decay rates for solutions of Maxwell's system with a memory boundary condition
We consider the stabilization of Maxwell's equations with space variable coefficients in a bounded region with a smooth boundary, subject to dissipative boundary conditions of memory type on the boundary. Under suitable conditions on the domain and on the permeability and permittivity coefficients, we prove the exponential/polynomial decay of the energy. Our result is mainly based on the use of the multipliers method and the introduction of a suitable Lyapounov functional.
Identifiability, stability and reconstruction results of point sources by boundary measurements in heteregeneous trees.
We consider the inverse problem of determining point wave sources in heteregeneous trees, extensions of one-dimensional stratified sets. We show that the Neumann boundary observation on a part of the lateral boundary determines uniquely the point sources if the time of observation is large enough. We further establish a conditional stability and give a reconstructing scheme.
Stabilization of a coupled multidimensional system.
We introduce a model of a vibrating multidimensional structure made of a n-dimensional body and a one-dimensional rod. We actually consider the anisotropic elastodynamic system in the n-dimensional body and the Euler-Bernouilli beam in the one-dimensional rod. These equations are coupled via their boundaries. Using appropriate feedbacks on a part of the boundary we show the exponential decay of the energy of the system.
Regularity of the solution of some transmission problems in domains with cuspidal point
Regularity results for transmission problems in domains with (outgoing) cuspidal points are considered. We prove in some special but generic situations that the solution is piecewise in .
On a coupled problem between the plate equation and the membrane equation on polygons
Control of networks of Euler-Bernoulli beams
We consider the exact controllability problem by boundary action of hyperbolic systems of networks of Euler-Bernoulli beams. Using the multiplier method and Ingham's inequality, we give sufficient conditions insuring the exact controllability for all time. These conditions are related to the spectral behaviour of the associated operator and are sufficiently concrete in order to be able to check them on particular networks as illustrated on simple examples.
error estimates for a nonconforming finite element discretization of the heat equation
The paper presents an error estimator for a (piecewise linear) nonconforming finite element approximation of the heat equation in , 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 with...
Boundary stabilization of Maxwell's equations with space-time variable coefficients
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. ...
Stabilization of second order evolution equations with unbounded feedback with delay
We consider abstract second order evolution equations with unbounded feedback with delay. Existence results are obtained under some realistic assumptions. Sufficient and explicit conditions are derived that guarantee the exponential or polynomial stability. Some new examples that enter into our abstract framework are presented.
Zienkiewicz–Zhu error estimators on anisotropic tetrahedral and triangular finite element meshes
We consider error estimators that can be applied to tetrahedral finite element meshes, 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 analyses of the estimators...
Discrete compactness for a discontinuous Galerkin approximation of Maxwell's system
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.
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