We consider a model problem (with constant coefficients and simplified geometry) for the boundary layer phenomena which appear in thin shell theoryas the relative thickness ε of the shell tends tozero. For ε = 0 our problem is parabolic, then it is amodel of developpable surfaces. Boundary layers along and across the characteristichave very different structure. It also appears internal layers associatedwith propagations of singularities along the characteristics. The specialstructure of the limit problem often implies solutions which exhibitdistributional singularities along the characteristics. The correspondinglayers for small ε have a very large intensity. Layers alongthe characteristics have a special structure involving subspaces; thecorresponding Lagrange multipliers are exhibited. Numerical experimentsshow the advantage of adaptive meshes in these problems.

A new numerical method based on fictitious domain methods for shapeoptimization problems governed by the Poisson equation is proposed.The basic idea is to combine the boundary variation technique, in whichthe mesh is moving during the optimization, and efficient fictitiousdomain preconditioning in the solution of the (adjoint) state equations.Neumann boundary value problems are solved using an algebraic fictitiousdomain method. A mixed formulation based on boundary Lagrangemultipliers is used for Dirichlet boundary problems and the resultingsaddle-point problems are preconditioned with block diagonal fictitiousdomain preconditioners. Under given assumptions on the meshes, thesepreconditioners are shown to be optimal with respect to the conditionnumber. The numerical experiments demonstrate the efficiency ofthe proposed approaches.

We consider a domain decomposition method for some unsteadyheat conduction problem in composite structures.This linear model problem is obtained by homogenization of thin layersof fibres embedded into some standard material.For ease of presentation we consider the case of two space dimensions only.The set of finite element equations obtained by the backward Euler schemeis parallelized in a problem-oriented fashion by some noniterative overlappingdomain splitting method,eventually enhanced by inexpensive local iterationsto reduce the overlap.We present a detailed convergence analysis of this algorithmwhich is particularly well appropriate to handle fibre layersof nonlinear material.Special emphasis is to take into account the specific regularity propertiesof the present mathematical model.Numerical experiments show the reliability of the theoretical predictions.

Finite element semidiscrete approximations on nonlinear dynamicshallow shell models in considered. It is shown that the algorithmleads to global, optimal rates of convergence. The resultpresented in the paper improves upon the existing literature where therates of convergence were derived for small initial data only[19].

In this paper, a general technique is developed to enlarge the velocity space ${\rm V}_h^1$ of the unstable -element by adding spaces ${\rm V}_h^2$ such thatfor the extended pair the Babuska-Brezzi condition is satisfied. Examplesof stable elements which can be derived in such a way imply the stability of the well-knownQ2/Q1 -element and the 4Q1/Q1 -element. However, our new elementsare much more cheaper. In particular, we shall see that more than half of theadditional degrees of freedom when switching from the Q 1 to the Q 2 and4Q1 , respectively, element are not necessary to stabilize theQ1/Q1 -element. Moreover, by using the technique of reduced discretizationsand eliminating the additional degrees of freedom we showthe relationship between enlarging the velocity space and stabilized methods. This relationship has been established for triangular elements but was not known for quadrilateral elements. As a result we derive new stabilizedmethods for the Stokes and Navier-Stokes equations. Finally, we showhow the Brezzi-Pitkäranta stabilization and the SUPG method for theincompressible Navier-Stokes equations can be recovered as special cases of the general approach. In contrast to earlier papers we do not restrict ourselves to linearized versions of the Navier-Stokes equations but deal with the full nonlinear case.

The aim of this work is to establish, from amathematical point of view, the limit α → +∞ in the system $i \partial_t E+\nabla (\nabla . E)-\alpha^2 \nabla \times\nabla \times E =-|E|^{2\sigma}E,$ where $E:{\ensuremath{{\Bbb R}}}^3\rightarrow{\mathbb C}^3$ . This corresponds to an approximationwhich is made in the context of Langmuir turbulence in plasmaPhysics. The L 2-subcritical σ (that is σ ≤ 2/3)and the H 1-subcritical σ (that is σ ≤ 2) arestudied. In the physical case σ = 1, the limit is then studied for the $H^1({\ensuremath{{\Bbb R}}}^3)$ norm.

In the shape from shading problem of computer vision oneattempts to recover the three-dimensional shape of an object orlandscape from the shading on a single image. Under theassumptions that the surface is dusty, distant, and illuminatedonly from above, the problem reduces to that of solving theeikonal equation |Du|=f on a domain in $\mathbb{R}^2$ . Despitevarious existence and uniqueness theorems for smooth solutions,we show that this problem is unstable, which is catastrophic forgeneral numerical algorithms.

We investigate sufficient and possiblynecessary conditions for the L 2 stability of the upwind first orderfinite volume scheme for Maxwell equations, with metallic andabsorbing boundary conditions. We yield a very general sufficient condition,valid for any finite volume partition in two and three spacedimensions. We show this condition is necessary for a class ofregular meshes in two space dimensions. However, numerical tests show it is not necessaryin three space dimensions even on regular meshes. Stability limits for time and space schemes with higher orders of accuracy are numericallyinvestigated.

We consider H(curl;Ω)-elliptic problems that have been discretized bymeans of Nédélec's edge elements on tetrahedral meshes. Suchproblems occur in the numerical computation of eddy currents. From the defectequation we derive localized expressions that can be used as a posteriori error estimators to control adaptiverefinement. Under certain assumptions on material parameters and computational domains, we derive local lower bounds and a global upper bound for the total error measured in the energy norm. The fundamental tool in the numerical analysis is a Helmholtz-type decomposition of the error into an irrotational part and a weakly solenoidal part.

This work is devoted to the study of a two-dimensional vectorPoisson equation with the normal component of the unknown andthe value of the divergence of the unknown prescribed simultaneouslyon the entire boundary.These two scalar boundary conditions appear prima faciealternative in a standard variational framework. An originalvariational formulation of this boundary value problemis proposed here. Furthermore, an uncoupled solution algorithm isintroduced together with its finite element approximation.The numerical scheme has been implemented and appliedto solve a simple test problem.