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.

Cultivating oleaginous microalgae in specific culturing devices such as raceways is seenas a future way to produce biofuel. The complexity of this process coupling non linearbiological activity to hydrodynamics makes the optimization problem very delicate. Thelarge amount of parameters to be taken into account paves the way for a usefulmathematical modeling. Due to the heterogeneity of raceways along the depth dimensionregarding temperature, light intensity or nutrients availability, we adopt a multilayerapproach for hydrodynamics and biology. For free surface hydrodynamics, we use amultilayer Saint–Venant model that allows mass exchanges, forced by a simplifiedrepresentation of the paddlewheel. Then, starting from an improved Droop model thatincludes light effect on algae growth, we derive a similar multilayer system for thebiological part. A kinetic interpretation of the whole system results in an efficientnumerical scheme. We show through numerical simulations in two dimensions that ourapproach is capable of discriminating between situations of mixed water or calm andheterogeneous pond. Moreover, we exhibit that a posteriori treatment ofour velocity fields can provide lagrangian trajectories which are of great interest toassess the actual light pattern perceived by the algal cells and therefore understand itsimpact on the photosynthesis process.

We analyze the regularity of random entropy solutions to scalar hyperbolic conservation laws with random initial data. We prove regularity theorems for statistics of random entropy solutions like expectation, variance, space-time correlation functions and polynomial moments such as gPC coefficients. We show how regularity of such moments (statistical and polynomial chaos) of random entropy solutions depends on the regularity of the distribution law of the random shock location of the initial data. Sufficient conditions on the law of the initial data for moments of the random entropy solution to be piece-wise smooth functions of space and time are identified, even in cases where path-wise random entropy solutions are discontinuous almost surely. We extrapolate the results to hyperbolic systems of conservation laws in one space dimension. We then exhibit a class of stochastic Galerkin discretizations which allows to derive closed deterministic systems of hyperbolic conservation laws for the coefficients in truncated polynomial chaos expansions of the random entropy solution. Based on the regularity theory developed here, we show that depending on the smoothness of the law of the initial data, arbitrarily high convergence rates are possible for the computation of coefficients in gPC approximations of random entropy solutions for Riemann problems with random shock location by combined Stochastic Galerkin Finite Volume schemes.

In this article, we present a numerical scheme based on a finite element method in orderto solve a time-dependent convection-diffusion equation problem and satisfy someconservation properties. In particular, our scheme is able to conserve the total energyfor a heat equation or the total mass of a solute in a fluid for a concentration equation,even if the approximation of the velocity field is not completely divergence-free. Weestablish a priori errror estimates for this scheme and we give some numerical exampleswhich show the efficiency of the method.

We prove the discrete compactness property of the edge elements of any order on a classof anisotropically refined meshes on polyhedral domains. The meshes, made up oftetrahedra, have been introduced in [Th. Apel and S. Nicaise, Math. Meth. Appl.Sci. 21 (1998) 519–549]. They are appropriately graded nearsingular corners and edges of the polyhedron.

In this paper, we first construct a model for free surface flows that takes into accountthe air entrainment by a system of four partial differential equations. We derive it bytaking averaged values of gas and fluid velocities on the cross surface flow in the Eulerequations (incompressible for the fluid and compressible for the gas). The obtained systemis conditionally hyperbolic. Then, we propose a mathematical kinetic interpretation ofthis system to finally construct a two-layer kinetic scheme in which a special treatmentfor the “missing” boundary condition is performed. Several numerical tests on closed waterpipes are performed and the impact of the loss of hyperbolicity is discussed andillustrated. Finally, we make a numerical study of the order of the kinetic method in thecase where the system is mainly non hyperbolic. This provides a useful stability resultwhen the spatial mesh size goes to zero.

We propose an unconditionally stable semi-implicit time discretization of the phase fieldcrystal evolution. It is based on splitting the underlying energy into convex and concaveparts and then performing H-1 gradient descent steps implicitly for the formerand explicitly for the latter. The splitting is effected in such a way that the resultingequations are linear in each time step and allow an extremely simple implementation andefficient solution. We provide the associated stability and error analysis as well asnumerical experiments to validate the method’s efficiency.

We consider optimal control problems for the bidomain equations of cardiacelectrophysiology together with two-variable ionic models, e.g. theRogers–McCulloch model. After ensuring the existence of global minimizers, we provide arigorous proof for the system of first-order necessary optimality conditions. The proof isbased on a stability estimate for the primal equations and an existence theorem for weaksolutions of the adjoint system.

In this work we study a fully discrete mixed scheme, based on continuous finite elementsin space and a linear semi-implicit first-order integration in time, approximating anEricksen–Leslie nematic liquid crystal model by means of aGinzburg–Landau penalized problem. Conditional stability of this schemeis proved via a discrete version of the energy law satisfied by thecontinuous problem, and conditional convergence towards generalized Young measure-valuedsolutions to the Ericksen–Leslie problem is showed when the discreteparameters (in time and space) and the penalty parameter go to zero at the same time.Finally, we will show some numerical experiences for a phenomenon of annihilation ofsingularities.

We discuss a numerical formulation for the cell problem related to a homogenization approach for the study of wetting on micro rough surfaces. Regularity properties of the solution are described in details and it is shown that the problem is a convex one. Stability of the solution with respect to small changes of the cell bottom surface allows for an estimate of the numerical error, at least in two dimensions. Several benchmark experiments are presented and the reliability of the numerical solution is assessed, whenever possible, by comparison with analytical one. Realistic three dimensional simulations confirm several interesting features of the solution, improving the classical models of study of wetting on roughness.

We review the optimal design of an arterial bypass graft following either a (i) boundaryoptimal control approach, or a (ii) shape optimization formulation. The main focus isquantifying and treating the uncertainty in the residual flow when the hosting artery isnot completely occluded, for which the worst-case in terms of recirculation effects isinferred to correspond to a strong orifice flow through near-complete occlusion.Aworst-case optimal control approach is applied to the steady Navier-Stokes equations in 2Dto identify an anastomosis angle and a cuffed shape that are robust with respect to apossible range of residual flows. We also consider a reduced order modelling frameworkbased on reduced basis methods in order to make the robust design problem computationallyfeasible. The results obtained in 2D are compared with simulations in a 3D geometry butwithout model reduction or the robust framework.

A weak solution of the coupling of time-dependent incompressible Navier–Stokes equationswith Darcy equations is defined. The interface conditions include theBeavers–Joseph–Saffman condition. Existence and uniqueness of the weak solution areobtained by a constructive approach. The analysis is valid for weak regularityinterfaces.

A new class of nonparametric nonconforming quadrilateral finite elements is introducedwhich has the midpoint continuity and the mean value continuity at the interfaces ofelements simultaneously as the rectangular DSSY element [J. Douglas, Jr., J.E. Santos, D.Sheen and X. Ye, ESAIM: M2AN 33 (1999) 747–770.] Theparametric DSSY element for general quadrilaterals requires five degrees of freedom tohave an optimal order of convergence [Z. Cai, J. Douglas, Jr., J.E. Santos, D. Sheen andX. Ye, Calcolo 37 (2000) 253–254.], while the newnonparametric DSSY elements require only four degrees of freedom. The design of newelements is based on the decomposition of a bilinear transform into a simple bilinear mapfollowed by a suitable affine map. Numerical results are presented to compare the newelements with the parametric DSSY element.

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.

Iterative approximation algorithms are successfully applied in parametric approximation tasks. In particular, reduced basis methods make use of the so-called Greedy algorithm for approximating solution sets of parametrized partial differential equations. Recently, a priori convergence rate statements for this algorithm have been given (Buffa et al. 2009, Binev et al. 2010). The goal of the current study is the extension to time-dependent problems, which are typically approximated using the POD–Greedy algorithm (Haasdonk and Ohlberger 2008). In this algorithm, each greedy step is invoking a temporal compression step by performing a proper orthogonal decomposition (POD). Using a suitable coefficient representation of the POD–Greedy algorithm, we show that the existing convergence rate results of the Greedy algorithm can be extended. In particular, exponential or algebraic convergence rates of the Kolmogorov n-widths are maintained by the POD–Greedy algorithm.

Existence of a solution to the quasi-variational inequality problem arising in a modelfor sand surface evolution has been an open problem for a long time. Another long-standingopen problem concerns determining the dual variable, the flux of sand pouring down theevolving sand surface, which is also of practical interest in a variety of applications ofthis model. Previously, these problems were solved for the special case in which theinequality is simply variational. Here, we introduce a regularized mixed formulationinvolving both the primal (sand surface) and dual (sand flux) variables. We derive,analyse and compare two methods for the approximation, and numerical solution, of thismixed problem. We prove subsequence convergence of both approximations, as the meshdiscretization parameters tend to zero; and hence prove existence of a solution to thismixed model and the associated regularized quasi-variational inequality problem. One ofthese numerical approximations, in which the flux is approximated by thedivergence-conforming lowest order Raviart–Thomas element, leads to an efficient algorithmto compute not only the evolving pile surface, but also the flux of pouring sand. Resultsof our numerical experiments confirm the validity of the regularization employed.

We consider the flow of a viscous incompressible fluid in a rigid homogeneous porousmedium provided with mixed boundary conditions. Since the boundary pressure can presenthigh variations, the permeability of the medium also depends on the pressure, so that themodel is nonlinear. A posteriori estimates allow us to omit thisdependence where the pressure does not vary too much. We perform the numerical analysis ofa spectral element discretization of the simplified model. Finally we propose a strategywhich leads to an automatic identification of the part of the domain where the simplifiedmodel can be used without increasing significantly the error.

Using Burgers’ equation with mixed Neumann–Dirichlet boundary conditions, we highlight aproblem that can arise in the numerical approximation of nonlinear dynamical systems oncomputers with a finite precision floating point number system. We describe the dynamicalsystem generated by Burgers’ equation with mixed boundary conditions, summarize some ofits properties and analyze the equilibrium states for finite dimensional dynamical systemsthat are generated by numerical approximations of this system. It is important to notethat there are two fundamental differences between Burgers’ equation with mixedNeumann–Dirichlet boundary conditions and Burgers’ equation with both Dirichlet boundaryconditions. First, Burgers’ equation with homogenous mixed boundary conditions on a finiteinterval cannot be linearized by the Cole–Hopf transformation. Thus, on finite intervalsBurgers’ equation with a homogenous Neumann boundary condition is truly nonlinear. Second,the nonlinear term in Burgers’ equation with a homogenous Neumann boundary condition isnot conservative. This structure plays a key role in understanding the complex dynamicsgenerated by Burgers’ equation with a Neumann boundary condition and how this structureimpacts numerical approximations. The key point is that, regardless of the particularnumerical scheme, finite precision arithmetic will always lead to numerically generatedequilibrium states that do not correspond to equilibrium states of the Burgers’ equation.In this paper we establish the existence and stability properties of these numericalstationary solutions and employ a bifurcation analysis to provide a detailed mathematicalexplanation of why numerical schemes fail to capture the correct asymptotic dynamics. Weextend the results in [E. Allen, J.A. Burns, D.S. Gilliam, J. Hill and V.I. Shubov,Math. Comput. Modelling 35 (2002) 1165–1195] and provethat the effect of finite precision arithmetic persists in generating a nonzero numericalfalse solution to the stationary Burgers’ problem. Thus, we show that the results obtainedin [E. Allen, J.A. Burns, D.S. Gilliam, J. Hill and V.I. Shubov, Math. Comput.Modelling 35 (2002) 1165–1195] are not dependent on a specifictime marching scheme, but are generic to all convergent numerical approximations ofBurgers’ equation.

We consider the following problem of error estimation for the optimal control ofnonlinear parabolic partial differential equations: let an arbitrary admissible controlfunction be given. How far is it from the next locally optimal control? Under naturalassumptions including a second-order sufficient optimality condition for the (unknown)locally optimal control, we estimate the distance between the two controls. To do this, weneed some information on the lowest eigenvalue of the reduced Hessian. We apply thistechnique to a model reduced optimal control problem obtained by proper orthogonaldecomposition (POD). The distance between a local solution of the reduced problem to alocal solution of the original problem is estimated.

We propose a new reduced basis element-cum-component mode synthesis approach forparametrized elliptic coercive partial differential equations. In the Offline stage weconstruct a Library of interoperable parametrized reference componentsrelevant to some family of problems; in the Online stage we instantiate andconnect reference components (at ports) to rapidly form and query parametricsystems. The method is based on static condensation at the interdomainlevel, a conforming eigenfunction “port” representation at the interface level, andfinally Reduced Basis (RB) approximation of Finite Element (FE) bubble functions at theintradomain level. We show under suitable hypotheses that the RB Schur complement is closeto the FE Schur complement: we can thus demonstrate the stability of the discreteequations; furthermore, we can develop inexpensive and rigorous (system-level) aposteriori error bounds. We present numerical results for model many-parameterheat transfer and elasticity problems with particular emphasis on the Online stage; wediscuss flexibility, accuracy, computational performance, and also the effectivity of thea posteriori error bounds.