We extend our results on fictitious domain methods for Poisson’s problem to the case of incompressible elasticity, or Stokes’ problem. The mesh is not fitted to the domain boundary. Instead boundary conditions are imposed using a stabilized Nitsche type approach. Control of the non-physical degrees of freedom, i.e., those outside the physical domain, is obtained thanks to a ghost penalty term for both velocities and pressures. Both inf-sup stable and stabilized velocity pressure pairs are considered.

We address multiscale elliptic problems with random coefficients that are a perturbationof multiscale deterministic problems. Our approach consists in taking benefit of theperturbative context to suitably modify the classical Finite Element basis into adeterministic multiscale Finite Element basis. The latter essentially shares the sameapproximation properties as a multiscale Finite Element basis directly generated on therandom problem. The specific reference method that we use is the Multiscale Finite ElementMethod. Using numerical experiments, we demonstrate the efficiency of our approach and thecomputational speed-up with respect to a more standard approach. In the stationarysetting, we provide a complete analysis of the approach, extending that available for thedeterministic periodic setting.

We consider an uncoupled, modular regularization algorithm for approximation of theNavier-Stokes equations. The method is: Step 1.1: Advance the NSEone time step, Step 1.1: Regularize to obtain the approximation atthe new time level. Previous analysis of this approach has been for specific time steppingmethods in Step 1.1 and simple stabilizations in Step 1.1. In this report we extend the mathematical support for uncoupled,modular stabilization to (i) the more complex and better performing BDF2 timediscretization in Step 1.1, and (ii) more general (linear ornonlinear) regularization operators in Step 1.1. We give a completestability analysis, derive conditions on the Step 1.1regularization operator for which the combination has good stabilization effects,characterize the numerical dissipation induced by Step 1.1, provean asymptotic error estimate incorporating the numerical error of the method used in Step1.1 and the regularizations consistency error in Step 1.1 and provide numerical tests.

In this article, we prove convergence of the weakly penalized adaptive discontinuousGalerkin methods. Unlike other works, we derive the contraction property for variousdiscontinuous Galerkin methods only assuming the stabilizing parameters are large enoughto stabilize the method. A central idea in the analysis is to construct an auxiliarysolution from the discontinuous Galerkin solution by a simple post processing. Based onthe auxiliary solution, we define the adaptive algorithm which guides to the convergenceof adaptive discontinuous Galerkin methods.

We present in this paper a proof of well-posedness and convergence for the parallelSchwarz Waveform Relaxation Algorithm adapted to an N-dimensional semilinear heatequation. Since the equation we study is an evolution one, each subproblem at each stephas its own local existence time, we then determine a common existence time for everyproblem in any subdomain at any step. We also introduce a new technique: Exponential DecayError Estimates, to prove the convergence of the Schwarz Methods, with multisubdomains,and then apply it to our problem.

This article aims at studying the controllability of a simplified fluid structureinteraction model derived and developed in [C. Conca, J. Planchard and M. Vanninathan,RAM: Res. Appl. Math. John Wiley & Sons Ltd., Chichester (1995);J.-P. Raymond and M. Vanninathan, ESAIM: COCV 11 (2005)180–203; M. Tucsnak and M. Vanninathan, Systems Control Lett. 58(2009) 547–552]. This interaction is modeled by a wave equation surrounding aharmonic oscillator. Our main result states that, in the radially symmetric case, thissystem can be controlled from the outer boundary. This improves previous results [J.-P.Raymond and M. Vanninathan, ESAIM: COCV 11 (2005) 180–203;M. Tucsnak and M. Vanninathan, Systems Control Lett. 58(2009) 547–552]. Our proof is based on a spherical harmonic decomposition of thesolution and the so-called lateral propagation of the energy for 1d waves.

In this note we show the characteristic function of every indecomposable setF in theplane is BVequivalent to the characteristic function a closed set $\mathrm{See\; Formula\; in\; PDF}$\hbox{See Formula in PDF} .We show by example this is false in dimension three and above. As a corollary to thisresult we show that for every ϵ > 0 a set of finite perimeter S can be approximated by aclosed subset $\mathrm{See\; Formula\; in\; PDF}$\hbox{See Formula in PDF} with finitely many indecomposablecomponents and with the property that $\mathrm{See\; Formula\; in\; PDF}$\hbox{See Formula in PDF} and $\mathrm{See\; Formula\; in\; PDF}$\hbox{See Formula in PDF} .We apply this corollary to give a short proof that locally quasiminimizing sets in theplane are BVlextension domains.

We consider the optimal distribution of several elastic materials in a fixed workingdomain. In order to optimize both the geometry and topology of the mixture we rely on thelevel set method for the description of the interfaces between the different phases. Wediscuss various approaches, based on Hadamard method of boundary variations, for computingshape derivatives which are the key ingredients for a steepest descent algorithm. Theshape gradient obtained for a sharp interface involves jump of discontinuous quantities atthe interface which are difficult to numerically evaluate. Therefore we suggest analternative smoothed interface approach which yields more convenient shape derivatives. Werely on the signed distance function and we enforce a fixed width of the transition layeraround the interface (a crucial property in order to avoid increasing “grey” regions offictitious materials). It turns out that the optimization of a diffuse interface has itsown interest in material science, for example to optimize functionally graded materials.Several 2-d examples of compliance minimization are numerically tested which allow us tocompare the shape derivatives obtained in the sharp or smoothed interface cases.

In this note we prove compactness for the Cahn–Hilliard functional without assumingcoercivity of the multi-well potential.

A dual-weighted residual approach for goal-oriented adaptive finite elements for a classof optimal control problems for elliptic variational inequalities is studied. Thedevelopment is based on the concept of C-stationarity. The overall error representationdepends on primal residuals weighted by approximate dual quantities and vice versaas well as various complementarity mismatch errors. Also, a prioribounds for C-stationary points and associated multipliers are derived. Details onthe numerical realization of the adaptive concept are provided and a report on numericaltests including the critical cases of biactivity are presented.

Problems of wave interaction with a body with arbitrary shape floating or submerged in water are of immense importance in the literature on the linearized theory of water waves. Wave-free potentials are used to construct solutions to these problems involving bodies with circular geometry, such as a submerged or half-immersed long horizontal circular cylinder (in two dimensions) or sphere (in three dimensions). These are singular solutions of Laplace’s equation satisfying the free surface condition and decaying rapidly away from the point of singularity. Wave-free potentials in two and three dimensions for infinitely deep water as well as water of uniform finite depth with a free surface are known in the literature. The method of constructing wave-free potentials in three dimensions is presented here in a systematic manner, neglecting or taking into account the effect of surface tension at the free surface or for water with an ice cover modelled as a thin elastic plate floating on the water. The forms of the wave motion at the upper surface (free surface or ice-covered surface) related to these wave-free potentials are depicted graphically in a number of figures for all the cases considered.