Faker Ben Belgacem

The Mortar finite element method with Lagrange multipliers

Summary. The present paper deals with a variant of a non conforming domain decomposition technique: the mortar finite element method. In the opposition to the original method this variant is never conforming because of the relaxation of the matching constraints at the vertices (and the edges in 3D) of subdomains. It is shown that, written under primal hybrid formulation, the approximation problem, issued from a discretization of a second order elliptic equation in 2D, is nonetheless well posed and provides a discrete solution that satisfies optimal error estimates with respect to natural norms. Finally the parallelization advantages consequence of this variant are also addressed.

The mortar finite element method for contact problems

The purpose of this paper is to describe a domain decomposition technique: the mortar finite element method applied to contact problems between two elastic bodies. This approach allows the use of no-matching grids and to glue different discretizations across the contact zone in an optimal way, at least for bilateral contact. We present also an adaptation of this method to unilateral contact problems.

EXTENSION OF THE MORTAR FINITE ELEMENT METHOD TO A VARIATIONAL INEQUALITY MODELING UNILATERAL CONTACT

The purpose of this paper is to extend the mortar finite element method to handle the unilateral contact model between two deformable bodies. The corresponding variational inequality is approximated using finite element meshes which do not fit on the contact zone. The mortar technique allows one to match these independent discretizations of each solid and takes into account the unilateral contact conditions in a convenient way. By using an adaptation of Falks lemma and a bootstrap argument, we give an upper bound of the convergence rate similar to the one already obtained for compatible meshes.

Why is the Cauchy problem severely ill-posed?

An answer to the ill-posedness degree issue of the Cauchy problem may be found in the theory of kernel operators. The foundation of the proof is the Steklov–Poincare approach introduced in Ben Belgacem and El Fekih (2005 Inverse Problems 21 1915–36), which consists of reformulating the Cauchy problem as a variational equation, in an appropriate Sobolev scale, and is set on the part of the boundary where data are missing. The linear (Steklov–Poincare) operator involved in that reduced problem turns out to be compact with a non-closed range; hence the ill-posedness. Conducting an accurate spectral analysis of this operator requires characterization of it as a kernel operator, which is obtained through Greens functions of the (Laplace) differential equation. The severe ill-posedness is then settled for smooth domains after showing a fast decaying towards zero of the eigenvalues of that Steklov–Poincare operator. This is achieved by applying the Weyl–Courant min–max principle and some polynomial approximation results. Addressing more general smooth domains with corners, we discuss the regularity of Greens function and we explain why there is a room to extend our analysis to this case and why we are optimistic that it will definitely establish the severe ill-posedness of the Cauchy problem.

On Cauchy's problem: I. A variational Steklov–Poincaré theory

In 1923 (Lectures on Cauchys Problem in Linear PDEs (New York, 1953)), J Hadamard considered a particular example to illustrate the ill-posedness of the Cauchy problem for elliptic partial differential equations, which consists in recovering data on the whole boundary of the domain from partial but over-determined measures. He achieved explicit computations for the Laplace operator, due to the squared shape of the domain, to observe, in fine, that the solution does not depend continuously on the given boundary data. The primary subject of this contribution is to extend the result to general domains by proving that the Cauchy problem has a variational formulation that can be put under a (variational) pseudo-differential equation, set on the boundary where the data are missing, and defined by a compact Steklov?Poincar?-type operator. The construction of this operator is based on the Dirichlet-to-Neumann mapping, and its compactness is derived from the elliptic regularity theory. Next, using mathematical tools from the linear operator theory and the convex optimization, we provide a comprehensive analysis of the reduced problem which enables us to state that (i) the set of compatible data, for which existence and uniqueness are guaranteed, is dense in the admissible data space; (ii) when the existence fails, due to possible noisy data, the variational problem can be consistently approximated by the least-squares method, that is the incompatibility measure (the deviation indicator or the variational crime made on the Steklov?Poincar? equation) equals zero though all the minimizing sequences blow up.

On Cauchy's problem: II. Completion, regularization and approximation

In Ben Belgacem and El Fekih (2005 On Cauchys problem: I. A variational Steklov–Poincare theory Inverse Problems 21 1915–36), a new variational theory is introduced for the data completion Cauchy problem, and studied in the Sobolev scales. Reformulating it, owing to the Dirichlet-to-Neumann operator, enables us to prove several mathematical results for the obtained Steklov–Poincare problem and to establish the connection with some well-known minimization methods. In particular, when the over-specified data are incompatible and the existence fails for the Cauchy problem, it is stated that the least-squares incompatibility measure equals zero and so does the minimum value of the Kohn–Vogelius function, though all the minimizing sequences blow up. Because of the ill-posedness of the Cauchy–Steklov–Poincare problem, an efficient numerical simulation of it can scarcely be achieved without some regularization materials. When combined with carefully chosen stopping criteria, they bring stability to the computations and dampen the noise perturbations caused by possibly erroneous measurements. This paper, part II, is the numerical counterpart of I and handles some practical issues. We are mainly involved in the Tikhonov scheme and the finite-element method applied to the unstable data completion problem. We lay down a new non-distributional space, in the Steklov–Poincare framework, that allows for an elegant investigation of the reliability of both regularizations in: (i) approximating the exact solution, for compatible data and (ii) providing a consistent pseudo-solution, for incompatible data, that turns also to be a minimizing sequence of the Kohn–Vogelius gap function. Moreover, some convergence estimates with respect to the regularization parameters are stated for some worthy indicators such as the incompatibility measure and the minimum value of the Kohn–Vogelius and energy functions. Finally, we report and discuss some informative computing experiences to support the theoretical predictions of each regularization and to assess their reliability.

The Mixed Mortar Finite Element Method for the Incompressible Stokes Problem: Convergence Analysis

Our purpose is to prove that the mortar element method, using local compatible mixed finite elements, provides optimal convergence results when applied to the incompressible Stokes equations.

Approximation du problème de contact unilatéral par la méthode des éléments finis avec joints

Abstract The purpose of this Note is to extend the mortar finite element method to handle the unilateral contact model between two deformable bodies. The corresponding variational inequality is approximated using finite elements with meshes which do not fit on the contact zone. The mortar technique allows us to match (independent) discretizations within each solid and to express the contact conditions in a satisfying way. Then, we carry out a numerical analysis of the algorithm and, using a bootstrap argument, we give an upper bound of the convergence rate similar to that already obtained for compatible grids.

Polynomial extensions of compatible polynomial traces in three dimensions

Abstract We describe the construction of polynomial extension operators of polynomial traces over the reference cube satisfying optimal stability conditions with respect to Sobolev norms. Combined with Hilbertian interpolation argument, these results show the equivalence of Sobolev norms and the interpolation norms on spaces of polynomials defined over a square.

Coupling Spectral and Finite Elements for Second Order Elliptic Three-Dimensional Equations

This paper deals with the approximation of elliptic second order problems with a nonconforming domain decomposition method that allows for coupling spectral and finite element discretizations in three dimensions. We present two kinds of matching conditions at the interfaces between the subdomains. The detailed numerical analysis of the error committed upon the exact solution turns out to be optimal. We describe also the implementation and give some numerical results confirming the theoretical predictions.