We first generalize, in an abstract framework, results on the order of convergence of a semi-discretization in time by an implicit Euler scheme of a stochastic parabolic equation. In this part, all the coefficients are globally Lipchitz. The case when the nonlinearity is only locally Lipchitz is then treated. For the sake of simplicity, we restrict our attention to the Burgers equation. We are not able in this case to compute a pathwise order of the approximation, we introduce the weaker notion of order in probability and generalize in that context the results of the globally Lipschitz case.

We introduce a new variational method for the numerical homogenization of divergence formelliptic, parabolic and hyperbolic equations with arbitrary rough (L∞)coefficients. Our method does not rely on concepts of ergodicity or scale-separation buton compactness properties of the solution space and a new variational approach tohomogenization. The approximation space is generated by an interpolation basis (overscattered points forming a mesh of resolution H) minimizing the L2 norm of thesource terms; its (pre-)computation involves minimizing 𝒪(H−d)quadratic (cell) problems on (super-)localized sub-domains of size 𝒪(H ln(1/H)).The resulting localized linear systems remain sparse and banded. The resultinginterpolation basis functions are biharmonic for d ≤ 3, and polyharmonic ford ≥ 4, forthe operator −div(a∇·) and can be seen as a generalization ofpolyharmonic splines to differential operators with arbitrary rough coefficients. Theaccuracy of the method (𝒪(H)in energy norm and independent from aspect ratios of the mesh formed by the scatteredpoints) is established via the introduction of a new class ofhigher-order Poincaré inequalities. The method bypasses (pre-)computations on the fulldomain and naturally generalizes to time dependent problems, it also provides a naturalsolution to the inverse problem of recovering the solution of a divergence form ellipticequation from a finite number of point measurements.

Semi–smooth Newton methods are analyzed for a class of variational inequalities in infinite dimensions.It is shown that they are equivalent to certain active set strategies. Global and local super-linear convergence areproved. To overcome the phenomenon of finite speed of propagation of discretized problems a penalty versionis used as the basis for a continuation procedure to speed up convergence. The choice of the penalty parameter can be made on the basis of an L∞ estimate for the penalized solutions. Unilateral as well as bilateral problems are considered.

We introduce a new second-order central-upwind scheme for the Saint-Venant system of shallow water equations on triangular grids. We prove thatthe scheme both preserves “lake at rest” steady states and guarantees the positivity of the computed fluid depth. Moreover, it can be appliedto models with discontinuous bottom topography and irregular channel widths. We demonstrate these features of the new scheme, as well as itshigh resolution and robustness in a number of numerical examples.

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.

Phase-field models, the simplest of which is Allen–Cahn's problem, are characterized by a small parameter ε that dictatesthe interface thickness. These models naturally call for mesh adaptation techniques, which rely on a posteriori error control. However, their error analysis usually deals with the underlying non-monotone nonlinearity via a Gronwall argument which leads to an exponential dependence on ε-2. Using an energy argument combined with a topological continuation argument and a spectral estimate, we establish an a posteriori error control result with only a low order polynomial dependence in ε-1. Our result is applicable to any conforming discretization technique that allows for a posteriori residual estimation. Residual estimators for an adaptive finite element scheme are derived to illustrate the theory.

The Discontinuous Galerkin Time Domain (DGTD) methods are now popular for the solution of wave propagation problems. Able to deal with unstructured, possibly locally-refined meshes, they handleeasily complex geometries and remain fully explicit with easy parallelization and extension to high orders of accuracy. Non-dissipative versions exist, where some discrete electromagnetic energy is exactly conserved. However, the stability limit of the methods, related to the smallest elements in the mesh, calls for the construction of local-time stepping algorithms. These schemes have already been developed for N-body mechanical problems and are known as symplectic schemes. They are applied here to DGTD methods on wave propagation problems.

We show that while Runge-Kutta methods cannot preserve polynomial invariants in general, they can preserve polynomials that are the energy invariant of canonical Hamiltonian systems.

The numerical solution of the elliptic Monge-Ampère Partial Differential Equation has been a subject of increasing interest recently [Glowinski, in 6th International Congress on Industrial and Applied Mathematics, ICIAM 07, Invited Lectures (2009) 155–192;Oliker and Prussner, Numer. Math.54 (1988) 271–293; Oberman, Discrete Contin. Dyn. Syst. Ser. B10 (2008) 221–238; Dean and Glowinski, in Partial differential equations, Comput. Methods Appl. Sci. 16 (2008) 43–63; Glowinski et al., Japan J. Indust. Appl. Math.25 (2008) 1–63; Dean and Glowinski, Electron. Trans. Numer. Anal.22 (2006) 71–96; Dean and Glowinski, Comput. Methods Appl. Mech. Engrg.195 (2006) 1344–1386; Dean et al., in Control and boundary analysis, Lect. Notes Pure Appl. Math. 240 (2005) 1–27; Feng and Neilan, SIAM J. Numer. Anal.47 (2009) 1226–1250; Feng and Neilan, J. Sci. Comput.38 (2009) 74–98; Feng and Neilan, http://arxiv.org/abs/0712.1240v1; G. Loeper and F. Rapetti, C. R. Math. Acad. Sci. Paris340 (2005) 319–324].There are already two methods available [Oliker and Prussner, Numer. Math.54 (1988) 271–293; Oberman, Discrete Contin. Dyn. Syst. Ser. B10 (2008) 221–238] which converge even for singular solutions.However, many of the newly proposed methods lack numerical evidence of convergence on singular solutions, or are known to break down in this case. In this article we present and study the performance of two methods.The first method, which is simply the natural finite difference discretization of the equation, is demonstrated to be the best performing method (in terms of convergence and solution time) currently available for generic (possibly singular) problems, in particular when the right hand side touches zero.The second method, which involves the iterative solution of a Poisson equation involving the Hessian of the solution, is demonstrated to be the best performing (in terms of solution time) when the solution is regular, which occurs when the right hand side is strictly positive.

The present work is a mathematical analysis of two algorithms, namelythe Roothaan and the level-shifting algorithms, commonly used inpractice to solve the Hartree-Fock equations. The level-shiftingalgorithm is proved to be well-posed and to converge provided the shiftparameter is large enough. On the contrary, cases when the Roothaanalgorithm is not well defined or fails in converging areexhibited. These mathematical results are confronted to numericalexperiments performed by chemists.

We consider the numerical solution of diffusion problems in (0,T) x Ω for $\Omega\subset \mathbb{R}^d$ and for T > 0 indimension dd ≥ 1. We use a wavelet based sparse gridspace discretization with mesh-width h and order pd ≥ 1, andhp discontinuous Galerkin time-discretization of order $r =O(\left|\log h\right|)$ on a geometric sequence of $O(\left|\log h\right|)$ many timesteps. The linear systems in each time step are solved iterativelyby $O(\left|\log h\right|)$ GMRES iterations with a wavelet preconditioner. We prove that this algorithm gives an L 2(Ω)-error of O(N-p) for u(x,T) where N is the total number of operations,provided that the initial data satisfies $u_0 \in H^\varepsilon(\Omega)$ with ε > 0 and that u(x,t) is smooth in x for t>0. Numerical experiments in dimension d up to 25 confirm thetheory.

The standard multilayer Saint-Venant system consists in introducing fluidlayers that are advected by the interfacial velocities. As a consequence there is no massexchanges between these layers and each layer is described by its height and its averagevelocity.Here we introduce another multilayer system with mass exchanges between the neighboringlayers where the unknowns are a total height of water and an average velocity per layer.We derive it from Navier-Stokes system with an hydrostatic pressure and prove energy andhyperbolicity properties of the model. We also give a kinetic interpretation leading toeffective numerical schemes with positivity and energy properties. Numerical tests showthe versatility of the approach and its ability to compute recirculation cases with windforcing.

Models of two phase flows in porous media, used in petroleumengineering, lead to a system of two coupled equations with ellipticand parabolic degenerate terms, and two unknowns,the saturation and the pressure.For the purpose of their approximation, a coupled scheme, consisting ina finite volume method together witha phase-by-phase upstream weighting scheme, is used in the industrial setting.This paper presents a mathematical analysis of this coupled scheme, first showingthat it satisfies some a priori estimates:the saturation is shown to remain in a fixed interval, anda discrete L2(0,T;H1 (Ω)) estimate is proved for both the pressureand a function of the saturation. Thanks to these properties,a subsequence of the sequence of approximate solutions is shown toconverge to a weak solutionof the continuous equationsas the size of the discretization tends to zero.

We study the gradient flow for the total variation functional, which arises in image processing and geometric applications. We propose a variational inequality weak formulation for the gradient flow,and establish well-posedness of the problem by the energy method. The main idea of our approach is to exploit the relationship betweenthe regularized gradient flow (characterized by a small positive parameterε, see (1.7)) and the minimal surface flow [21]and the prescribed mean curvature flow [16].Since our approach is constructive and variational, finite element methods can be naturally applied to approximate weak solutions of the limiting gradient flow problem. We propose a fully discrete finite element method and establish convergence tothe regularized gradient flow problem as h,k → 0, and to the total variation gradient flow problem as h,k,ε → 0in general cases.Provided that the regularized gradient flow problem possessesstrong solutions, which is proved possible if the datum functionsare regular enough, we establish practical a priori error estimates for the fully discrete finite element solution, in particular, by focusing on the dependence of the error bounds on the regularization parameter ε. Optimal order error bounds are derived for the numerical solution under the meshrelation k = O(h2). In particular, it is shown thatall error bounds depend on $\frac{1}{\varepsilon}$ onlyin some lower polynomial order for small ε.

Some electromagnetic materials have, in a given frequency range, an effective dielectric permittivity and/or a magnetic permeability which are real-valued negative coefficients when dissipation is neglected. They are usually called metamaterials. We study a scalar transmission problem between a classical dielectric material and a metamaterial, set in an open, bounded subset of Rd, with d = 2,3. Our aim is to characterize occurences where the problem is well-posed within the Fredholm (or coercive + compact) framework. For that, we build some criteria, based on the geometry of the interface between the dielectric and the metamaterial. The proofs combine simple geometrical arguments with the approach of T-coercivity, introduced by the first and third authors and co-worker. Furthermore, the use of localization techniques allows us to derive well-posedness under conditions that involve the knowledge of the coefficients only near the interface. When the coefficients are piecewise constant, we establish the optimality of the criteria.

We consider the coupling between three-dimensional(3D) and one-dimensional (1D) fluid-structure interaction (FSI) models describing blood flow inside compliant vessels. The 1D model is a hyperbolicsystem of partial differential equations.The 3D model consists of the Navier-Stokes equationsfor incompressible Newtonian fluids coupled with a model for the vessel wall dynamics. A non standard formulationfor the Navier-Stokes equations is adopted tohave suitable boundary conditions for the couplingof the models. With this we derive an energy estimatefor the fully 3D-1D FSI coupling. We consider several possiblemodels for the mechanics of the vessel wall in the 3D problemand show how the 3D-1D coupling depends on them.Several comparative numerical tests illustrating the coupling are presented.

We establish an asymptotic representation formula for the steady state voltageperturbations caused by low volume fraction internal conductivityinhomogeneities. This formula generalizes and unifies earlierformulas derived for special geometries and distributions of inhomogeneities.

The purpose of the present article is to compare different phase-spacesampling methods, such as purely stochastic methods (Rejection method, Metropolized independence sampler, Importance Sampling),stochastically perturbed Molecular Dynamics methods (Hybrid Monte Carlo, Langevin Dynamics, Biased Random Walk), and purelydeterministic methods (Nosé-Hoover chains, Nosé-Poincaré and RecursiveMultiple Thermostats (RMT) methods). After recalling some theoretical convergence properties forthe various methods, we provide some new convergence resultsfor the Hybrid Monte Carlo scheme, requiring weaker (and easier tocheck) conditions than previously known conditions. We then turn to the numericalefficiency of the sampling schemes for a benchmark model of linearalkane molecules. In particular, the numerical distributions that are generated are compared in a systematic way, on the basisof some quantitative convergence indicators.

The numerical approximation of parametric partial differential equations is acomputational challenge, in particular when the number of involved parameter is large.This paper considers a model class of second order, linear, parametric, elliptic PDEs on abounded domain D with diffusion coefficients depending on the parametersin an affine manner. For such models, it was shown in [9, 10] that under very weak assumptionson the diffusion coefficients, the entire family of solutions to such equations can besimultaneously approximated in the Hilbert space V = H01(D) by multivariate sparse polynomials in the parametervector y with a controlled number N of terms. Theconvergence rate in terms of N does not depend on the number ofparameters in V, which may be arbitrarily large or countably infinite,thereby breaking the curse of dimensionality. However, these approximation results do notdescribe the concrete construction of these polynomial expansions, and should thereforerather be viewed as benchmark for the convergence analysis of numerical methods. Thepresent paper presents an adaptive numerical algorithm for constructing a sequence ofsparse polynomials that is proved to converge toward the solution with the optimalbenchmark rate. Numerical experiments are presented in large parameter dimension, whichconfirm the effectiveness of the adaptive approach.

We consider a Vlasov-Fokker-Planck equation governing the evolution of the density of interacting and diffusive matter in the space of positions and velocities. We use a probabilistic interpretation to obtain convergence towards equilibrium in Wasserstein distance with an explicit exponential rate. We also prove a propagation of chaos property for an associated particle system, and give rates on the approximation of the solution by the particle system. Finally, a transportation inequality for the distribution of the particle system leads to quantitative deviation bounds on the approximation of the equilibrium solution of the equation by an empirical mean of the particles at given time.