Zakaria Belhachmi

Weighted Clément operator and application to the finite element discretization of the axisymmetric Stokes problem

We consider the Stokes problem in an axisymmetric three-dimensional domain with data which are axisymmetric and have angular component equal to zero. We observe that the solution is also axisymmetric and the velocity has also zero angular component, hence the solution satisfies a system of equations in the meridian domain. The weak three-dimensional problem reduces to a two-dimensional one with weighted integrals. The latter is discretized by Taylor–Hood type finite elements. A weighted Clément operator is defined and approximation results are proved. This operator is then used to derive the discrete inf–sup condition and optimal a priori error estimates.

How to Choose Interpolation Data in Images

We introduce and discuss shape-based models for finding the best interpolation data when reconstructing missing regions in images by means of solving the Laplace equation. The shape analysis is done in the framework of

The homogenization method for topology and shape optimization. Single and multiple loads case

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A direct method for photoacoustic tomography with inhomogeneous sound speed

-convergence, from two different points of view. First, we propose a continuous PDE model and get pointwise information on the “importance” of each pixel by a topological asymptotic method. Second, we introduce a finite dimensional setting into the continuous model based on fat pixels (balls with positive radius) and study by

Stability and Uniqueness for the Crack Identification Problem

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Spectral Element Discretization of the Circular Driven Cavity, Part II: The Bilaplacian Equation

-convergence the asymptotics when the radius vanishes. In this way, we obtain relevant information about the optimal distribution of the best interpolation pixels. We show that the resulting optimal data sets are identical to sets that can also be motivated using level set ideas and approximation theoretic considerations. Numerical computations are presented that confirm the usefulness of our theoretical findings for PDE-based image compression.

A TRUNCATED FOURIER/FINITE ELEMENT DISCRETIZATION OF THE STOKES EQUATIONS IN AN AXISYMMETRIC DOMAIN

ABSTRACT This paper is devoted to an elementary introduction to the homogenization methods applied to topology and shape optimization of elastic structures under single and multiple external loads. The single load case, in the context of minimum compliance and weight design of elastic structures, has been fully described in its theoretical as well as its numerical aspects in [4]. It is here briefly recalled. In the more realistic context of “multiple loads”, i.e. when the structure is optimized with respect to more than one set of external forces, most of the obtained theoretical results remain true. However, the parameters that define optimal composite materials cannot be computed explicity. In this paper, a method to treat numerically the multiple loads case is proposed.

Control of the Effects of Regularization on Variational Optic Flow Computations

The standard approach for photoacoustic imaging with variable speed of sound is time reversal, which consists in solving a well-posed final-boundary value problem for the wave equation backwards in time. This paper investigates the iterative Landweber regularization algorithm, where convergence is guaranteed by standard regularization theory, notably also in cases of trapping sound speed or for short measurement times. We formulate and solve the direct and inverse problem on the whole Euclidean space, what is common in standard photoacoustic imaging, but not for time-reversal algorithms, where the problems are considered on a domain enclosed by the measurement devices. We formulate both the direct and adjoint photoacoustic operator as the solution of an interior and an exterior differential equation which are coupled by transmission conditions. The prior is solved numerically using a Galerkin scheme in space and finite difference discretization in time, while the latter consists in solving a boundary integral equation. We therefore use a BEM-FEM approach for numerical solution of the forward operators. We analyze this method, prove convergence, and provide numerical tests. Moreover, we compare the approach to time reversal.

Numerical simulations of free convection about a vertical flat plate embedded in a porous medium

This paper deals with the identifiability of nonsmooth defects by boundary measurements, and the stability of their detection. We introduce and analyze a new pointwise regularity concept at the boundary of an open set which turns out to play a crucial role in the identifiability of defects by two boundary measurements. As a consequence, we prove the unique identifiability for a large class of closed sets, including sets with an infinite number of connected components of positive capacity and totally disconnected sets. In order to rigorously justify numerical approximation results of defects by optimal design methods, we prove a geometric stability result of the defect identification process, without any a priori smoothness assumptions.

On a Decomposition Model for Optical Flow

This paper is devoted to the spectral element discretization of the Laplace equation in a disk when provided with discontinuous boundary data. Relying on an appropriate variational formulation, we propose a discrete problem and prove its convergence. The use of weighted Sobolev spaces to treat the discontinuity of the boundary conditions also allows for improving the order of convergence. The results of the numerical experiments we present are in agreement with the theoretical ones.