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.

A discontinuous Galerkin finite element method for an optimalcontrol problem related to semilinear parabolic PDE's is examined.The schemes under consideration are discontinuous in time butconforming in space. Convergence of discrete schemes of arbitraryorder is proven. In addition, the convergence of discontinuousGalerkin approximations of the associated optimality system to thesolutions of the continuous optimality system is shown. The proofis based on stability estimates at arbitrary time points underminimal regularity assumptions, and a discrete compactnessargument for discontinuous Galerkin schemes (see Walkington[SINUM (June 2008) (submitted), preprint available at http://www.math.cmu.edu/~noelw], Sects. 3, 4).

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.

A new class of history-dependent quasivariational inequalities was recently studied in[M. Sofonea and A. Matei, History-dependent quasivariational inequalities arising incontact mechanics. Eur. J. Appl. Math. 22 (2011) 471–491].Existence, uniqueness and regularity results were proved and used in the study of severalmathematical models which describe the contact between a deformable body and an obstacle.The aim of this paper is to provide numerical analysis of the quasivariationalinequalities introduced in the aforementioned paper. To this end we introduce temporallysemi-discrete and fully discrete schemes for the numerical approximation of theinequalities, show their unique solvability, and derive error estimates. We then applythese results to a quasistatic frictional contact problem in which the material’s behavioris modeled with a viscoelastic constitutive law, the contact is bilateral, and friction isdescribed with a slip-rate version of Coulomb’s law. We discuss implementation of thecorresponding fully-discrete scheme and present numerical simulation results on atwo-dimensional example.

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.

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.

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.

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 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.

We study here some asymptotic models for the propagation of internal and surface waves in a two-fluid system. We focus on the so-called long wave regime for one-dimensional waves, and consider the case of a flat bottom. Following the method presented in [J.L. Bona, T. Colin and D. Lannes,Arch. Ration. Mech. Anal. 178 (2005) 373–410] for the one-layer case, we introduce a new family of symmetric hyperbolic models, that are equivalent to the classical Boussinesq/Boussinesq system displayed in [W. Choi and R. Camassa, J. Fluid Mech. 313 (1996) 83–103]. We study the well-posedness of such systems, and the asymptotic convergence of their solutions towards solutions of the full Euler system. Then, we provide a rigorous justification of the so-called KdV approximation, stating that any bounded solution of the full Euler system can be decomposed into four propagating waves, each of them being well approximated by the solutions of uncoupled Korteweg-de Vries equations. Our method also applies for models with the rigid lid assumption, using the Boussinesq/Boussinesq models introduced in [J.L. Bona, D. Lannes and J.-C. Saut, J. Math. Pures Appl. 89 (2008) 538–566]. Our explicit and simultaneous decomposition allows to study in details the behavior of the flow depending on the depth and density ratios, for both the rigid lid and free surface configurations. In particular, we consider the influence of the rigid lid assumption on the evolution of the interface, and specify its domain of validity. Finally, solutions of the Boussinesq/Boussinesq systems and the KdV approximation are numerically computed, using a Crank-Nicholson scheme with a predictive step inspired from [C. Besse,C. R. Acad. Sci. Paris Sér. I Math. 326 (1998) 1427–1432;C. Besse and C.H. Bruneau,Math. Mod. Methods Appl. Sci. 8 (1998) 1363–1386].

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.

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.

In this paper we introduce numerical schemes for aone-dimensional kinetic model of the Boltzmann equation withdissipative collisions and variable coefficient of restitution. Inparticular, we study the numerical passage of the Boltzmannequation with singular kernel to nonlinear friction equations inthe so-called quasi elastic limit. To this aim we introduce aFourier spectral method for the Boltzmann equation [CITE]and show that the kernel modes that define the spectral methodhave the correct quasi elastic limit providing a consistentspectral method for the limiting nonlinear friction equation.

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.

This paper is devoted to the numerical solution of stationarylaminar Bingham fluids by path-following methods. By using duality theory, asystem that characterizes the solution of the original problem is derived.Since this system is ill-posed, a family of regularized problems is obtainedand the convergence of the regularized solutions to the original one is proved.For the update of the regularization parameter, a path-following method isinvestigated. Based on the differentiability properties of the path, a model ofthe value functional and a correspondent algorithm are constructed. For thesolution of the systems obtained in each path-following iteration a semismoothNewton method is proposed. Numerical experiments are performed in order toinvestigate the behavior and efficiency of the method, and a comparison with apenalty-Newton-Uzawa-conjugate gradient method, proposed in [Dean et al., J. Non-Newtonian Fluid Mech.142 (2007) 36–62], iscarried out.

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.

The plane wave stability properties of the conservative schemes of Besse [SIAM J. Numer. Anal.42 (2004) 934–952] and Fei et al. [Appl. Math. Comput.71 (1995) 165–177] for the cubic Schrödinger equation are analysed. Although the two methods possess many of the same conservation properties, we show that their stability behaviour is very different.An energy preserving generalisation of the Fei method with improved stability is presented.