We give results for the approximation of a laminate withvarying volume fractions for multi-well energy minimizationproblems modeling martensitic crystals thatcan undergo either an orthorhombicto monoclinic or a cubic to tetragonal transformation. We construct energy minimizing sequences of deformations which satisfythe corresponding boundary condition, and we establish a series of error bounds in terms of the elastic energyfor the approximation of the limiting macroscopicdeformation and the simply laminated microstructure.Finally, we give results for the corresponding finite element approximation of the laminate with varying volume fractions.

In the present work, the symmetrized sequential-parallel decomposition method with the fourth order accuracy for the solution of Cauchy abstract problem with an operator under a split form is presented. The fourth order accuracy is reached by introducing a complex coefficient with the positive real part. For the considered scheme, the explicit a priori estimate is obtained.

We consider models based on conservation laws. For the optimizationof such systems, a sensitivity analysis is essential to determinehow changes in the decision variables influence the objectivefunction. Here we study the sensitivity with respect to the initialdata of objective functions that depend upon the solution of Riemannproblems with piecewise linear flux functions. We presentrepresentations for the one–sided directional derivatives of theobjective functions. The results can be used in the numerical methodcalled Front-Tracking.

A hyperelastic constitutive law, for use in anatomically accurate finite element models ofliving structures, is suggested for the passive and the active mechanical properties of incompressiblebiological tissues. This law considers the passive and active states as a same hyperelastic continuummedium, and uses an activation function in order to describe the whole contraction phase.The variational and the FE formulations are also presented, and the FE code has been validatedand applied to describe the biomechanical behavior of a thick-walled anisotropic cylinder underdifferent active loading conditions.

We propose and analyse a abstract framework for augmented mixed formulations.We give a priori error estimate in the general case: conforming and nonconforming approximations with or without numerical integration.Finally, a posteriori error estimator is given. An example of stabilized formulation for Stokes problem is analysed.

In this paper, we investigate the coupling between operator splitting techniques and a timeparallelization scheme, the parareal algorithm,as a numericalstrategy for the simulation of reaction-diffusion equations modelling multi-scale reaction waves. This type of problems induces peculiar difficulties and potentially large stiffness which stem from the broad spectrum of temporal scales in the nonlinear chemical source term as well as from the presence of large spatial gradients in the reactive fronts,spatially very localized.In a series of previous studies, the numerical analysis of theoperator splitting as well as the parareal algorithmhas been conducted and such approaches have shown a great potential in the framework of reaction-diffusion and convection-diffusion-reaction systems. However, complementary studies are needed for a more complete characterizationof such techniques for these stiff configurations.Therefore,we conduct in this work a precise numerical analysis that considers thecombination of time operator splitting and the parareal algorithmin the context of stiff reaction fronts. The impact of the stiffness featured by these frontson the convergence of the method is thus quantified, and allows to conclude on an optimal strategy for the resolution ofsuch problems.We finally perform some numerical simulations in the field of nonlinear chemical dynamics that validate the theoretical estimatesand examine the performance of such strategiesin the context of academical one-dimensional test casesas well as multi-dimensional configurationssimulated on parallel architecture.

In this paper, a Dirichlet-Neumann substructuring domaindecomposition method is presented for a finite elementapproximation to the nonlinear Navier-Stokes equations. It isshown that the Dirichlet-Neumann domain decomposition sequenceconverges geometrically to the true solution provided the Reynoldsnumber is sufficiently small. In this method, subdomain problemsare linear. Other version where the subdomain problems are linearStokes problems is also presented.

The chemical master equation is a fundamental equation in chemical kinetics. It underliesthe classical reaction-rate equations and takes stochastic effects into account. In thispaper we give a simple argument showing that the solutions of a large class of chemicalmaster equations are bounded in weighted ℓ1-spaces and possess high-ordermoments. This class includes all equations in which no reactions between two or morealready present molecules and further external reactants occur that add mass to thesystem. As an illustration for the implications of this kind of regularity, we analyze theeffect of truncating the state space. This leads to an error analysis for the finite stateprojections of the chemical master equation, an approximation that forms the basis of manynumerical methods.

We prove error estimates for the ultra weak variational formulation (UWVF) in 3D linearelasticity. We show that the UWVF of Navier’s equation can be derived as an upwinddiscontinuous Galerkin method. Using this observation, error estimates are investigatedapplying techniques from the theory of discontinuous Galerkin methods. In particular, wederive a basic error estimate for the UWVF in a discontinuous Galerkin type norm and thenan error estimate in the L2(Ω) norm in terms of the bestapproximation error. Our final result is an L2(Ω) norm errorestimate using approximation properties of plane waves to give an estimate for the orderof convergence. Numerical examples are presented.

In the present work the symmetrized sequential-parallel decomposition methodof the third degree precision for the solution of Cauchy abstract problemwith an operator under a split form, is presented. The third degreeprecision is reached by introducing a complex coefficient with the positivereal part. For the considered schema the explicit a priori estimation isobtained.

From Kantorovich’s theory we present a semilocal convergence result for Newton’s methodwhich is based mainly on a modification of the condition required to the second derivativeof the operator involved. In particular, instead of requiring that the second derivativeis bounded, we demand that it is centered. As a consequence, we obtain a modification ofthe starting points for Newton’s method. We illustrate this study with applications tononlinear integral equations of mixed Hammerstein type.

Using the min-plus version of the spectral radius formula, one proves: 1) that the unique eigenvalue of a min-plus eigenvalue problem depends continuously on parameters involved in the kernel defining the problem; 2) that the numerical method introduced by Chou and Griffiths to compute this eigenvalue converges.A toolbox recently developed at I.n.r.i.a. helps to illustrate these results. Frenkel-Kontorova models serve as example. The analogy with homogenization of Hamilton-Jacobi equations is emphasized.

We present a numerical algorithm to solve the micromagnetic equations based on tangential-plane minimization for the magnetization update and a homothethic-layer decomposition of outer space for the computation of the demagnetization field. As a first application, detailed results on the flower-vortex transition in the cube of Micromagnetic Standard Problem number 3 are obtained, which confirm, with a different method, those already present in the literature, and validate our method and code. We then turn to switching of small cubic or almost-cubic particles, in the single-domain limit. Our data show systematic deviations from the Stoner-Wohlfarth model due to the non-ellipsoidal shape of the particle, and in particular a non-monotone dependence on the particle size.

In this paper we consider the numerical computation of the optimal costfunction associated to the problem that consists in finding the minimum ofthe maximum of a scalar functional on a trajectory. We present anapproximation method for the numerical solution which employs bothdiscretization on time and on spatial variables. In this way, we obtain afully discrete problem that has unique solution. We give an optimal estimatefor the error between the approximated solution and the optimal costfunction of the original problem. Also, numerical examples are presented.

We present in this paper a pressure correction scheme for the drift-flux model combining finite element and finite volume discretizations, which is shown to enjoy essential stability features of the continuous problem: the scheme is conservative, the unknowns are kept within their physical bounds and, in the homogeneous case (i.e. when the drift velocity vanishes), the discrete entropy of the system decreases; in addition, when using for the drift velocity a closure law which takes the form of a Darcy-like relation, the drift term becomes dissipative.Finally, the present algorithm preserves a constant pressure and a constant velocity through moving interfaces between phases.To ensure the stability as well as to obtain this latter property, a key ingredient is to couple the mass balance and the transport equation for the dispersed phase in an original pressure correction step.The existence of a solution to each step of the algorithm is proven; in particular, the existence of a solution to the pressure correction step is derived as a consequence of a more general existence result for discrete problems associated to the drift-flux model.Numerical tests show a near-first-order convergence rate for the scheme, both in time and space, and confirm its stability.

In this work we design a new domain decomposition method for the Euler equations in 2 dimensions. The starting point is the equivalence with a third order scalar equation to whom we can apply an algorithm inspired from the Robin-Robin preconditioner for the convection-diffusion equation [Achdou and Nataf,C. R. Acad. Sci. Paris Sér. I325 (1997) 1211–1216]. Afterwards we translate it into an algorithm for the initial system and prove that at the continuous level and for a decomposition into 2 sub-domains, it converges in 2 iterations. This property cannot be conserved strictly at discrete level and for arbitrary domain decompositions but we still have numerical results which confirm a very good stability with respect to the various parameters of the problem (mesh size, Mach number, ...).

We introduce and analyze a fully-mixed finite element method for a fluid-solidinteraction problem in 2D. The model consists of an elastic body which is subject to agiven incident wave that travels in the fluid surrounding it. Actually, the fluid issupposed to occupy an annular region, and hence a Robin boundary condition imitating thebehavior of the scattered field at infinity is imposed on its exterior boundary, which islocated far from the obstacle. The media are governed by the elastodynamic and acousticequations in time-harmonic regime, respectively, and the transmission conditions are givenby the equilibrium of forces and the equality of the corresponding normal displacements.We first apply dual-mixed approaches in both domains, and then employ the governingequations to eliminate the displacement u of the solid and the pressure pof the fluid. In addition, since both transmission conditions become essential, they areenforced weakly by means of two suitable Lagrange multipliers. As a consequence, theCauchy stress tensor and the rotation of the solid, together with the gradient ofp and the traces of u and p on the boundary of thefluid, constitute the unknowns of the coupled problem. Next, we show that suitabledecompositions of the spaces to which the stress and the gradient of pbelong, allow the application of the Babuška–Brezzi theory and the Fredholm alternativefor analyzing the solvability of the resulting continuous formulation. The unknowns of thesolid and the fluid are then approximated by a conforming Galerkin scheme defined in termsof PEERS elements in the solid, Raviart–Thomas of lowest order in the fluid, andcontinuous piecewise linear functions on the boundary. Then, the analysis of the discretemethod relies on a stable decomposition of the corresponding finite element spaces andalso on a classical result on projection methods for Fredholm operators of index zero.Finally, some numerical results illustrating the theory are presented.

This paper introduces a scheme for the numerical approximation of a model for two turbulent flows with coupling at an interface. We consider the variational formulation of the coupled model, where the turbulent kinetic energy equation is formulated by transposition. We prove the convergence of the approximation to this formulation for 3D flows for large turbulent viscosities and smooth enough flows, whenever bounded in W 1,p Sobolev norms for p large enough. Under the same assumptions, we show that the limit is a solution of the initial problem. Finally, we give some numerical experiments to enlighten the theoretical work.

The propagation of the action potential in the heart chambers is accurately described bythe Bidomain model, which is commonly accepted and used in the specialistic literature.However, its mathematical structure of a degenerate parabolic system entails highcomputational costs in the numerical solution of the associated linear system. Domaindecomposition methods are a natural way to reduce computational costs, and OptimizedSchwarz Methods have proven in the recent years their effectiveness in accelerating theconvergence of such algorithms. The latter are based on interface matching conditions moreefficient than the classical Dirichlet or Neumann ones. In this paper we analyze anOptimized Schwarz approach for the numerical solution of the Bidomain problem. We assessthe convergence of the iterative method by means of Fourier analysis, and we investigatethe parameter optimization in the interface conditions. Numerical results in 2D and 3D aregiven to show the effectiveness of the method.

In this paper we test the feasibility of coupling two heterogeneous mathematical modeling integrated within two different codes residing on distant sites. A prototype is developed using Schwarz type domain decomposition as the mathematical tool for coupling. The computing technology for coupling uses a CORBA environment to implement a distributed client-server programming model. Domain decomposition methods are well suited to reducing complex physical phenomena into a sequence of parallel subproblems in time and space. The whole process is easily tuned to underlying hardware requirements.