Serge Nicaise

Stability and Instability Results of the Wave Equation with a Delay Term in the Boundary or Internal Feedbacks

In this paper we consider, in a bounded and smooth domain, the wave equation with a delay term in the boundary condition. We also consider the wave equation with a delayed velocity term and mixed Dirichlet-Neumann boundary condition. In both cases, under suitable assumptions, we prove exponential stability of the solution. These results are obtained by introducing suitable energies and by using some observability inequalities. If one of the above assumptions is not satisfied, some instability results are also given by constructing some sequences of delays for which the energy of some solutions does not tend to zero.

The finite element method with anisotropic mesh grading for elliptic problems in domains with corners and edges

This paper is concerned with a specific finite element strategy for solving elliptic boundary value problems in domains with corners and edges. First, the anisotropic singular behaviour of the solution is described. Then the finite element method with anisotropic, graded meshes and piecewise linear shape functions is investigated for such problems; the schemes exhibit optimal convergence rates with decreasing mesh size. For the proof, new local interpolation error estimates for functions from anisotropically weighted spaces are derived. Finally, a numerical experiment is described, that shows a good agreement of the calculated approximation orders with the theoretically predicted ones.

Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedbacks

In this paper we consider the wave equation on 1-d networks with a delay term in the boundary and/or transmission conditions. We first show the well posedness of the problem and the decay of an appropriate energy. We give a necessary and sufficient condition that guarantees the decay to zero of the energy. We further give sufficient conditions that lead to exponential or polynomial stability of the solution. Some examples are also given.

Coefficients of the singularities for elliptic boundary value problems on domains with conical points. III: finite element methods on polygonal domains

In the two first parts of this work [RAIRO Model. Math. Anal. Numer., 24 (1990), pp. 27e52], [RAIRO Model. Math. Anal. Numer., 24 (1990), pp. 343–367] formulas giving the coefficients arising in the singular expansion of the solutions of elliptic boundary value problems on nonsmooth domains are investigated. Now, for the case of homogeneous strongly elliptic operators with constant coefficients on polygonal domains, the solution of such problems by the finite element method is considered. In order to approximate the solution or the coefficients, different methods are used based on the expressions of the coefficients that were obtained in the first two parts; the dual singular function method is also generalized.

Crouzeix-Raviart type finite elements on anisotropic meshes

Summary. The paper deals with a non-conforming finite element method on a class of anisotropic meshes. The Crouzeix-Raviart element is used on triangles and tetrahedra. For rectangles and prismatic (pentahedral) elements a novel set of trial functions is proposed. Anisotropic local interpolation error estimates are derived for all these types of element and for functions from classical and weighted Sobolev spaces. The consistency error is estimated for a general differential equation under weak regularity assumptions. As a particular application, an example is investigated where anisotropic finite element meshes are appropriate, namely the Poisson problem in domains with edges. A numerical test is described.

The eigenvalue problem for networks of beams

Abstract In this paper, we consider the spectral analysis of different models of networks of Euler–Bernoulli beams. We first give the characteristic equation for the spectrum. Secondly, in some particular situations, we show that the spectrum depends only on the structure of the graph. Thirdly, we investigate the asymptotic behaviour of the eigenvalues by proving the so-called Weyls formula.

ANALYTIC REGULARITY FOR LINEAR ELLIPTIC SYSTEMS IN POLYGONS AND POLYHEDRA

We prove weighted anisotropic analytic estimates for solutions of second order elliptic boundary value problems in polyhedra. The weighted analytic classes which we use are the same as those introduced by Guo in 1993 in view of establishing exponential convergence for hp finite element methods in polyhedra. We first give a simple proof of the known weighted analytic regularity in a polygon, relying on a new formulation of elliptic a priori estimates in smooth domains with analytic control of derivatives. The technique is based on dyadic partitions near the corners. This technique can successfully be extended to polyhedra, providing isotropic analytic regularity. This is not optimal, because it does not take advantage of the full regularity along the edges. We combine it with a nested open set technique to obtain the desired three-dimensional anisotropic analytic regularity result. Our proofs are global and do not require the analysis of singular functions.

A POSTERIORI ERROR ESTIMATION FOR THE STOKES PROBLEM: ANISOTROPIC AND ISOTROPIC DISCRETIZATIONS

The paper presents a posteriori error estimators for the stationary Stokes problem. We consider anisotropic finite element discretizations (i.e. elements with very large aspect ratio) where conventional, isotropic error estimators fail. Our analysis covers two- and three-dimensional domains, conforming and non-conforming discretizations as well as different elements. This large variety of settings requires different approaches and results in different estimators. Furthermore many examples of finite element pairs that are covered by the analysis are presented. Lower and upper error bounds form the main result with minimal assumptions on the elements. The lower error bound is uniform with respect to the mesh anisotropy with the exception of nonconforming 3D discretizations made of pentahedra or hexahedra. The upper error bound depends on a proper alignment of the anisotropy of the mesh which is a common feature of anisotropic error estimation. In the special case of isotropic meshes, the results simplify, and upper and lower error bounds hold unconditionally. Some of the corresponding results seem to be novel (in particular for 3D domains), and cover element pairs of practical importance. The numerical experiments confirm the theoretical predictions and show the usefulness of the anisotropic error estimators.

Feedback boundary stabilization of wave equations with interior delay

In this paper we consider a boundary stabilization problem for the wave equation with interior delay. We prove an exponential stability result under some Lions geometric condition. The proof of the main result is based on an identity with multipliers that allows us to obtain a uniform decay estimate for a suitable Lyapunov functional.

Exact Boundary Controllability of Maxwell's Equations in Heterogeneous Media and an Application to an Inverse Source Problem

We examine the question of control of Maxwells equations in a heterogeneous medium with a nonsmooth boundary by means of control currents on the boundary of that medium. This requires the introduction and analysis of some functions spaces. Some energy estimates are established which allow us to get the control results owing to the Hilbert uniqueness method. We finally give an application to an inverse source problem.