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Numerical solution of an inverse initial boundary value problem for the wave equation in the presence of conductivity imperfections of small volume

Mark Asch, Marion Darbas, Jean-Baptiste Duval (2011)

ESAIM: Control, Optimisation and Calculus of Variations

We consider the numerical solution, in two- and three-dimensional bounded domains, of the inverse problem for identifying the location of small-volume, conductivity imperfections in a medium with homogeneous background. A dynamic approach, based on the wave equation, permits us to treat the important case of “limited-view” data. Our numerical algorithm is based on the coupling of a finite element solution of the wave equation, an exact controllability method and finally a Fourier inversion for...

Numerical solution of second order one-dimensional linear hyperbolic equation using trigonometric wavelets

Mahmood Jokar, Mehrdad Lakestani (2012)

Kybernetika

A numerical technique is presented for the solution of second order one dimensional linear hyperbolic equation. This method uses the trigonometric wavelets. The method consists of expanding the required approximate solution as the elements of trigonometric wavelets. Using the operational matrix of derivative, we reduce the problem to a set of algebraic linear equations. Some numerical example is included to demonstrate the validity and applicability of the technique. The method produces very accurate...

Numerical study of acoustic multiperforated plates

Abderrahmane Bendali, M’Barek Fares, Sophie Laurens, Sébastien Tordeux (2012)

ESAIM: Proceedings

It is rather classical to model multiperforated plates by approximate impedance boundary conditions. In this article we would like to compare an instance of such boundary conditions obtained through a matched asymptotic expansions technique to direct numerical computations based on a boundary element formulation in the case of linear acoustic.

Observability inequalities and measurable sets

Jone Apraiz, Luis Escauriaza, Gengsheng Wang, C. Zhang (2014)

Journal of the European Mathematical Society

This paper presents two observability inequalities for the heat equation over Ω × ( 0 , T ) . In the first one, the observation is from a subset of positive measure in Ω × ( 0 , T ) , while in the second, the observation is from a subset of positive surface measure on ∂ Ω × ( 0 , T ) . It also proves the Lebeau-Robbiano spectral inequality when Ω is a bounded Lipschitz and locally star-shaped domain. Some applications for the above-mentioned observability inequalities are provided.

Observability properties of a semi-discrete 1d wave equation derived from a mixed finite element method on nonuniform meshes

Sylvain Ervedoza (2010)

ESAIM: Control, Optimisation and Calculus of Variations

The goal of this article is to analyze the observability properties for a space semi-discrete approximation scheme derived from a mixed finite element method of the 1d wave equation on nonuniform meshes. More precisely, we prove that observability properties hold uniformly with respect to the mesh-size under some assumptions, which, roughly, measures the lack of uniformity of the meshes, thus extending the work [Castro and Micu, Numer. Math.102 (2006) 413–462] to nonuniform meshes. Our results...

Obstacle problems for scalar conservation laws

Laurent Levi (2001)

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

In this paper we are interested in bilateral obstacle problems for quasilinear scalar conservation laws associated with Dirichlet boundary conditions. Firstly, we provide a suitable entropy formulation which ensures uniqueness. Then, we justify the existence of a solution through the method of penalization and by referring to the notion of entropy process solution due to specific properties of bounded sequences in L ∞ . Lastly, we study the behaviour of this solution and its stability properties with...

Obstacle problems for scalar conservation laws

Laurent Levi (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

In this paper we are interested in bilateral obstacle problems for quasilinear scalar conservation laws associated with Dirichlet boundary conditions. Firstly, we provide a suitable entropy formulation which ensures uniqueness. Then, we justify the existence of a solution through the method of penalization and by referring to the notion of entropy process solution due to specific properties of bounded sequences in L∞. Lastly, we study the behaviour of this solution and its stability properties...

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