We discuss the multi-configurationtime-dependent Hartree (MCTDH) method for the approximation of the time-dependent Schrödinger equation in quantum molecular dynamics. This method approximates the high-dimensional nuclearwave function by a linear combination of products of functions dependingonly on a single degree of freedom. The equations of motion, obtained via the Dirac-Frenkel time-dependent variational principle, consist of a coupled system of low-dimensionalnonlinear partial differential equations and ordinary differential equations. We show that, with a smooth and bounded potential, the MCTDH equations are well-posed and retain high-order Sobolev regularity globally in time, that is,as long as the density matrices appearing in the method formulation remain invertible. In particular, the solutions are regular enough to ensure local quasi-optimality of the approximation and to admit an efficient numerical treatment.

With the pioneering work of [Pardoux and Peng, Syst. Contr. Lett.14 (1990) 55–61; Pardoux and Peng,Lecture Notes in Control and Information Sciences176 (1992) 200–217]. We have at our disposalstochastic processes which solve the so-called backward stochasticdifferential equations. These processes provide us with a Feynman-Kacrepresentation for the solutions of a class of nonlinear partial differential equations (PDEs) which appearin many applications in the field of Mathematical Finance. Therefore thereis a great interest among both practitioners and theoreticians to developreliable numerical methods for their numerical resolution. In this survey, we present a number of probabilistic methods forapproximating solutions of semilinear PDEs all based on the correspondingFeynman-Kac representation. We also include a general introduction tobackward stochastic differential equations and their connection with PDEsand provide a generic framework that accommodates existing probabilisticalgorithms and facilitates the construction of new ones.

The purpose of this article is the analysis and the development of Eulerian multi-fluid models to describe the evolution of the mass density of evaporating liquid sprays.First, the classical multi-fluid model developed in [Laurent and Massot, Combust. Theor. Model.5 (2001) 537–572] is analyzed in the framework of an unsteady configuration without dynamical nor heating effects, where the evaporation process is isolated, since it is a key issue.The classical multi-fluid method consists then in a discretization of the droplet size variable into cells called sections.This analysis provides a justification of the “right” choice for this discretization to obtain a first order accurate and monotone scheme, with no restrictive CFL condition.This result leads to the development of a class of methods of arbitrary high order accuracy through the use of moments on the droplet surface in each section and a Godunov type method.Moreover, an extension of the two moments method is proposed which preserves the positivity and limits the total variation.Numerical results of the multi-fluid methods are compared to examine their capability to accurately describe the mass density in the spray with a small number of variables.This is shown to be a key point for the use of such methods in realistic flow configurations.

A current procedure that takes into account the Dirichlet boundary conditionwith non-smooth data is to change it into aRobin type condition by introducing a penalization term; a major effect of thisprocedure is an easy implementation of the boundary condition. In this work, we deal with an optimal control problem wherethe control variable is the Dirichlet data.We describe the Robin penalization,and we bound the gap between the penalized and the non-penalized boundary controlsfor the small penalization parameter. Some numerical results are reported on to highlightthe reliability of such an approach.

In this paper we elaborate a model to describe some aspects of the human lung considered as a continuous, deformable, medium. To that purpose, westudy the asymptotic behavior of aspring-mass system with dissipation. The key feature of our approach is the nature of this dissipation phenomena, which is related here to the flow of a viscous fluid through a dyadic tree of pipes (the branches), each exit of which being connected to an air pocket (alvelola) delimited by two successive masses.The first part focuses on the relation between fluxes and pressures at the outlets of a dyadic tree, assuming the flow within the tree obeys Poiseuille-like laws. In a second part, which contains the main convergence result, we intertwine the outlets of the tree with a spring-mass array. Letting again the number of generations (and therefore the number of masses) go to infinity, we show that the solutions to the finite dimensional problems converge in a weak sense to the solution of a wave-like partial differential equation with a non-local dissipative term.

The standard discretization of the Stokes andNavier–Stokes equations in vorticity and stream function formulation by affine finiteelements is known for its bad convergence. We present here a modified discretization, weprove that the convergence is improved and we establish a priori error estimates.

We analyze residual and hierarchicala posteriori error estimates for nonconforming finite elementapproximations of elliptic problems with variable coefficients.We consider a finite volume box scheme equivalent toa nonconforming mixed finite element method in a Petrov–Galerkinsetting. We prove thatall the estimators yield global upper and local lower bounds for the discretizationerror. Finally, we present results illustrating the efficiency of theestimators, for instance, in the simulation of Darcy flows throughheterogeneous porous media.

We consider the efficient and reliable solution of linear-quadratic optimal controlproblems governed by parametrized parabolic partial differential equations. To this end,we employ the reduced basis method as a low-dimensional surrogate model to solve theoptimal control problem and develop a posteriori error estimationprocedures that provide rigorous bounds for the error in the optimal control and theassociated cost functional. We show that our approach can be applied to problems involvingcontrol constraints and that, even in the presence of control constraints, the reducedorder optimal control problem and the proposed bounds can be efficiently evaluated in anoffline-online computational procedure. We also propose two greedy sampling procedures toconstruct the reduced basis space. Numerical results are presented to confirm the validityof our approach.

For a nonconforming finite element approximation of an elliptic model problem, we propose a posteriori error estimates in the energy norm which use as an additive term the “post-processing error” between the original nonconforming finite element solution and an easy computable conforming approximation of that solution.Thus, for the error analysis, the existing theory from the conformingcase can be used together with some simple additional arguments.As an essential point, the property is exploited that the nonconformingfinite element space contains as a subspace a conforming finite element space of first order. This property is fulfilled for many knownnonconforming spaces. We prove local lower and global upper a posteriori error estimates for an enhanced error measure which is the discretization error in the discrete energy norm plus the error of the best representation of the exact solution by a function in the conforming space used for thepost-processing. We demonstrate that the idea to use a computed conforming approximation ofthe nonconforming solution can be applied also to derive an a posteriorierror estimate for a linear functional of the solution which representssome quantity of physical interest.

An optimal control problem governed by a bilinear elliptic equation is considered. Thisproblem is solved by the sequential quadratic programming (SQP) method in aninfinite-dimensional framework. In each level of this iterative method the solution oflinear-quadratic subproblem is computed by a Galerkin projection using proper orthogonaldecomposition (POD). Thus, an approximate (inexact) solution of the subproblem isdetermined. Based on a POD a-posteriori error estimator developed byTröltzsch and Volkwein [Comput. Opt. Appl. 44 (2009) 83–115]the difference of the suboptimal to the (unknown) optimal solution of the linear-quadraticsubproblem is estimated. Hence, the inexactness of the discrete solution is controlled insuch a way that locally superlinear or even quadratic rate of convergence of the SQP isensured. Numerical examples illustrate the efficiency for the proposed approach.

We consider an initial and Dirichlet boundary value problem fora fourth-order linear stochastic parabolic equation, in one spacedimension, forced by an additive space-time white noise.Discretizing the space-time white noise a modelling error isintroduced and a regularized fourth-order linear stochasticparabolic problem is obtained. Fully-discrete approximations to the solution of the regularizedproblem are constructed by using, for discretization in space, aGalerkin finite element method based on C0 or C1piecewise polynomials, and, for time-stepping, the Backward Eulermethod.We derive strong a priori estimates for the modelling error and forthe approximation error to the solution of the regularizedproblem.

Many numerical simulations in (bilinear) quantum control use the monotonically convergent Krotov algorithms (introduced byTannor et al. [Time Dependent Quantum Molecular Dynamics (1992) 347–360]), Zhu and Rabitz [J. Chem. Phys. (1998) 385–391] or theirunified form described in Maday and Turinici [J. Chem. Phys. (2003) 8191–8196]. InMaday et al. [Num. Math. (2006) 323–338], a time discretization which preserves theproperty of monotonicity has been presented. This paper introduces aproof of the convergence of these schemes and some results regarding theirrate of convergence.

A new approach for computationally efficient estimation of stability factors forparametric partial differential equations is presented. The general parametric bilinearform of the problem is approximated by two affinely parametrized bilinear forms atdifferent levels of accuracy (after an empirical interpolation procedure). The successiveconstraint method is applied on the coarse level to obtain a lower bound for the stabilityfactors, and this bound is extended to the fine level by adding a proper correction term.Because the approximate problems are affine, an efficient offline/online computationalscheme can be developed for the certified solution (error bounds and stability factors) ofthe parametric equations considered. We experiment with different correction terms suitedfor a posteriori error estimation of the reduced basis solution ofelliptic coercive and noncoercive problems.

We present a reduced basis offline/online procedure for viscous Burgers initial boundaryvalue problem, enabling efficient approximate computation of the solutions of thisequation for parametrized viscosity and initial and boundary value data. This procedurecomes with a fast-evaluated rigorous error bound certifying the approximation procedure.Our numerical experiments show significant computational savings, as well as efficiency ofthe error bound.

We prove the convergence of a finite volumemethod for a noncoercive linear elliptic problem, with right-handside in the dual space of the natural energy space of the problem.

We present a high-resolution, non-oscillatory semi-discrete central scheme for one-dimensional shallow-water flows along channels with non uniform cross sections of arbitrary shape and bottom topography. The proposed scheme extends existing central semi-discrete schemes for hyperbolic conservation laws and enjoys two properties crucial for the accurate simulation of shallow-water flows: it preserves the positivity of the water height, and it is well balanced, i.e., the source terms arising from the geometry of the channel are discretized so as to balance the non-linear hyperbolic flux gradients. In addition to these, a modification in the numerical flux and the estimate of the speed of propagation, the scheme incorporates the ability to detect and resolve partially wet regions, i.e., wet-dry states. Along with a detailed description of the scheme and proofs of its properties, we present several numerical experiments that demonstrate the robustness of the numerical algorithm.

The Schwarz alternating method can be used to solveelliptic boundary value problems on domains which consist of two or moreoverlapping subdomains. The solution is approximated by an infinite sequence offunctions which results from solving a sequence of elliptic boundary value problems in each of the subdomains. In this paper, proofs of convergence of some Schwarz alternating methods fornonlinear elliptic problems which are known to have solutions by the monotonemethod (also known as the method of subsolutions and supersolutions) aregiven. In particular, an additive Schwarz method for scalar as well as somecoupled nonlinear PDEs are shown to converge for finitely many subdomains.These results are applicable to several models in population biology.

We rigorously establish the existence of the limithomogeneous constitutive law of a piezoelectric composite made of periodicallyperforated microstructures and whose reference configuration is athin shell with fixed thickness. We deal with an extension of theKoiter shell model in which the three curvilinear coordinates ofthe elastic displacement field and the electric potential arecoupled. By letting the size of themicrostructure going to zero and by using the periodicunfolding method combined with the Korn's inequality in perforateddomains, we obtain the limit model.

We study a two-grid scheme fully discrete in time andspace for solving the Navier-Stokes system. In the first step, thefully non-linear problem is discretized in space on a coarse gridwith mesh-size H and time step k. In the second step, theproblem is discretized in space on a fine grid with mesh-size hand the same time step, and linearized around the velocity u H computed in the first step. The two-grid strategy is motivated bythe fact that under suitable assumptions, the contribution ofu H to the error in the non-linear term, is measured in theL 2 norm in space and time, and thus has a higher-order than ifit were measured in the H 1 norm in space. We present thefollowing results: if h = H2 = k, then the global error ofthe two-grid algorithm is of the order of h, the same as wouldhave been obtained if the non-linear problem had been solveddirectly on thefine grid.

In this paper we consider a model shape optimization problem. The state variable solvesan elliptic equation on a domain with one part of the boundary described as the graph of acontrol function. We prove higher regularity of the control and develop a priorierror analysis for the finite element discretization of the shape optimizationproblem under consideration. The derived a priori error estimates areillustrated on two numerical examples.