We solve an optimal cost problem for a stochasticNavier-Stokes equation in spacedimension 2 by proving existence and uniqueness of a smooth solution of the corresponding Hamilton-Jacobi-Bellman equation.

The reduced basis method is a model reduction technique yielding substantial savings ofcomputational time when a solution to a parametrized equation has to be computed for manyvalues of the parameter. Certification of the approximation is possible by means of ana posteriori error bound. Under appropriate assumptions, this errorbound is computed with an algorithm of complexity independent of the size of the fullproblem. In practice, the evaluation of the error bound can become very sensitive toround-off errors. We propose herein an explanation of this fact. A first remedy has beenproposed in [F. Casenave, Accurate a posteriori error evaluation in thereduced basis method. C. R. Math. Acad. Sci. Paris 350(2012) 539–542.]. Herein, we improve this remedy by proposing a new approximationof the error bound using the empirical interpolation method (EIM). This method achieveshigher levels of accuracy and requires potentially less precomputations than the usualformula. A version of the EIM stabilized with respect to round-off errors is also derived.The method is illustrated on a simple one-dimensional diffusion problem and athree-dimensional acoustic scattering problem solved by a boundary element method.

In this paper, several modifications of the quasi-interpolationoperator of Scott andZhang [30]are discussed. The modified operators are defined for non-smooth functionsand are suited for application on anisotropic meshes. Theanisotropy of the elements is reflected in the local stability andapproximation error estimates.As an application, an example is considered where anisotropic finiteelement meshes are appropriate, namely the Poisson problem in domainswith edges.

We study a time-delay regularization of the anisotropic diffusion model for image denoising of Perona and Malik [IEEE Trans. Pattern Anal. Mach. Intell12 (1990) 629–639], which has been proposed by Nitzberg and Shiota [IEEE Trans. Pattern Anal. Mach. Intell14 (1998) 826–835]. In the two-dimensional case, we show the convergence of a numerical approximation and the existence of a weak solution. Finally, we show some experiments on images.

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 a class of discrete convex functionals which satisfy a (generalized) coarea formula. These functionals, based on submodular interactions, arise in discrete optimization and are known as a large class of problems which can be solved in polynomial time. In particular, some of them can be solved very efficiently by maximal flow algorithms and are quite popular in the image processing community. We study the limit in the continuum of these functionals, show that they always converge to some “crystalline” perimeter/total variation, and provide an almost explicit formula for the limiting functional.

Error estimates in L∞(0,T;L 2(Ω)), L∞(0,T;L 2(Ω)2),L∞(0,T;L ∞(Ω)), L∞(0,T;L∞ (Ω)2), Ω in ${\mathbb R}^2$ , are derived for a mixed finite element method for the initial-boundary value problem for integro-differential equation $$u_t={\rm div}\{a\bigtriangledown u+\int^t_0b_1\bigtriangledown u{\rm d}\tau+\int^t_0{\bf c}u{\rm d}\tau\}+f$$ based on the Raviart-Thomas space V h x Wh ⊂ H(div;Ω) x L2(Ω). Optimal order estimates are obtained for theapproximation of u,ut in L∞(0,T;L 2(Ω)) and the associated velocity p in L∞(0,T;L 2(Ω)2), divp in L∞(0,T;L 2(Ω)). Quasi-optimal order estimates are obtained for the approximation of u in L∞ (0,T;L ∞(Ω)) and p in L∞(0,T;L ∞(Ω)2.

We consider here the Interior Penalty Discontinuous Galerkin (IPDG) discretization of the wave equation. We show how to derive the optimal penalization parameter involved in this method in the case of regular meshes. Moreover, we provide necessary stability conditions of the global scheme when IPDG is coupled with the classical Leap–Frog scheme for the time discretization. Numerical experiments illustrate the fact that these conditions are also sufficient.

In long-time numerical integration of Hamiltonian systems,and especially in molecular dynamics simulation,it is important that the energy is well conserved. For symplecticintegrators applied with sufficiently small step size, thisis guaranteed by the existence of a modifiedHamiltonian that is exactly conserved up to exponentially smallterms. This article is concerned with the simplifiedTakahashi-Imada method, which is a modificationof the Störmer-Verlet method that is as easy to implement buthas improved accuracy. This integrator is symmetric andvolume-preserving, but no longer symplectic. We study itslong-time energy conservation and give theoreticalarguments, supported by numerical experiments, whichshow the possibility of a drift in the energy (linear or like a random walk).With respect to energy conservation, this article provides empiricaland theoretical data concerning the importance of using a symplecticintegrator.

In this paper we outline the hyperbolic system of governing equationsdescribing one-dimensional blood flow in arterial networks. Thissystem is numerically discretised using a discontinuous Galerkinformulation with a spectral/hp element spatial approximation. Weapply the numerical model to arterial networks in theplacenta. Starting with a single placenta we investigate the velocity waveformin the umbilical artery and its relationship with the distalbifurcation geometry and the terminal resistance. We then present resultsfor the waveform patterns and the volume fluxes throughout a simplifiedmodel of the arterial placental network in a monochorionic twin pregnancy with an arterio-arterial anastomosis and an arterio-venous anastomosis. Theeffects of varying the time period of the two fetus' heart beats, increasing the input flux of one fetus and the role ofterminal resistance in the network are investigated and discussed. The results show that the main features of the in vivo, physiological waves are captured by thecomputational model and demonstrate the applicability of themethods to the simulation of flows in arterial networks.

We are concerned with a 2D time harmonic wave propagationproblem in a medium including a thin slot whose thickness εis small with respect to the wavelength. In a previous article, we derivedformally an asymptotic expansion of the solution with respect to εusing the method of matched asymptotic expansions. We also proved theexistence and uniqueness of the terms of the asymptotics. In this paper,we complete the mathematical justification of our work by deriving optimal error estimates between the exact solutions and truncated expansions at any order.

We present and analyze an interior penalty method for the numerical discretization of the indefinite time-harmonic Maxwell equations in mixed form. The method is based on the mixed discretization of the curl-curl operator developed in [Houston et al., J. Sci. Comp.22 (2005) 325–356] and can be understood as a non-stabilized variant of the approach proposed in [Perugia et al., Comput. Methods Appl. Mech. Engrg.191 (2002) 4675–4697]. We show the well-posedness of this approach and derive optimal a priori error estimates in the energy-norm as well as the L2-norm. The theoretical results are confirmed in a series of numerical experiments.

We derive a posteriori estimates for a discretization in space of the standardCahn–Hilliard equation with a double obstacle free energy.The derived estimates are robust and efficient, and in practice are combinedwith a heuristic time step adaptation. We present numerical experiments in two and three space dimensions and compareour method with an existing heuristic spatial mesh adaptation algorithm.

We extend the classical empirical interpolation method [M. Barrault, Y. Maday, N.C.Nguyen and A.T. Patera, An empirical interpolation method: application to efficientreduced-basis discretization of partial differential equations. Compt. Rend. Math.Anal. Num. 339 (2004) 667–672] to a weighted empiricalinterpolation method in order to approximate nonlinear parametric functions with weightedparameters, e.g. random variables obeying various probabilitydistributions. A priori convergence analysis is provided for the proposedmethod and the error bound by Kolmogorov N-width is improved from the recent work [Y.Maday, N.C. Nguyen, A.T. Patera and G.S.H. Pau, A general, multipurpose interpolationprocedure: the magic points. Commun. Pure Appl. Anal. 8(2009) 383–404]. We apply our method to geometric Brownian motion, exponentialKarhunen–Loève expansion and reduced basis approximation of non-affine stochastic ellipticequations. We demonstrate its improved accuracy and efficiency over the empiricalinterpolation method, as well as sparse grid stochastic collocation method.

We consider the following reaction-diffusion equation:

$$ {\rm (KS)}\left\{\begin{array}{llll}u_t=\nabla \cdot \Big( \nabla u^m - u^{q-1} \nabla v \Big),& x \in \mathbb{R}^N, \ 0<t<\infty, \nonumber0 = \Delta v - v + u, & x \in \mathbb{R}^N, \ 0<t<\infty, \nonumberu(x,0) = u_0(x), & x \in \mathbb{R}^N,\end{array}\right.$$

where $N \ge 1, \ m > 1, \ q \ge \max\{m+\frac{2}{N},2\}$ .
In [Sugiyama, Nonlinear Anal.63 (2005) 1051–1062; Submitted; J. Differential Equations (in press)]it was shown that in the case of $q \ge \max\{m+\frac{2}{N},2\}$ , the above problem (KS) is solvable globally in time for “small $L^{\frac{N(q-m)}{2}}$ data”.Moreover, the decay of the solution (u,v) in $L^p(\mathbb{R}^N)$ was proved.In this paper, we consider the case of “ $q \ge \max\{m+\frac{2}{N},2\}$ and small $L^{\ell}$ data” with any fixed $\ell \ge \frac{N(q-m)}{2}$ and show that (i) there exists a time global solution (u,v) of (KS) andit decays to 0 as t tends to ∞ and(ii) a solution u of the first equation in (KS)behaveslike the Barenblatt solution asymptotically as t tends to ∞,where the Barenblatt solution is the exact solution (with self-similarity) of the porous medium equation $u_t = \Delta u^m$ with m>1.

We show that it is possible to construct a class of entropicschemes for the multicomponent Euler system describing a gas or fluidhomogeneous mixture at thermodynamic equilibrium by applying a relaxation technique. Afirst order Chapman–Enskog expansion shows that the relaxed systemformally converges when the relaxation frequencies go to the infinitytoward a multicomponent Navier–Stokes system with the classical Fick andNewton laws, with a thermal diffusion which can be assimilated to a Soret effect in the case of a fluid mixture,and with also a pressure diffusion or a density diffusion respectively for a gas or fluid mixture. We also discuss on the link between the convexity of the entropies of each species and the existence of the Chapman–Enskog expansion.

Linear Force-free (or Beltrami) fields are three-components divergence-free fields solutions of the equation curlB = α B, where α is a real number.Such fields appear in many branches of physics like astrophysics, fluid mechanics, electromagnetics and plasma physics. In this paper,we deal with some related boundary value problems in multiply-connected bounded domains, in half-cylindrical domains and in exterior domains.

The idea of replacing a divergence constraint by a divergence boundarycondition is investigated. The connections between the formulations areconsidered in detail. It is shown that the most common methods of usingdivergence boundary conditions do not always work properly. Necessaryand sufficient conditions for the equivalence of the formulations aregiven.

We consider a body immersed in a perfect gas and moving under the action of a constant force.Body and gas are in thermal equilibrium. We assume a stochastic interaction body/medium: when a particle of the medium hits the body, it is absorbed and immediately re-emitted with a Maxwellian distribution. This system gives rise to a microscopic model of friction. We study the approach of the body velocity V(t) to the limiting velocity $V_\infty$ and prove that, under suitable smallness assumptions, the approach to equilibrium is $$|V(t)-V_\infty|\approx \frac{C}{t^{d+1}},$$ where d is the dimension of the space, and C is a positive constant. This approach is not exponential, as typical in friction problems, and even slower than for the same problem with elastic collisions.

We discuss best N-term approximation spaces for one-electron wavefunctions $\phi_i$ and reduced density matrices ρemerging from Hartree-Fock and density functional theory. The approximation spaces $A^\alpha_q(H^1)$ for anisotropicwavelet tensor product bases have been recently characterized by Nitsche in terms of tensor product Besov spaces. We have used the norm equivalence of these spaces to weighted $\ell_q$ spaces of wavelet coefficients toproof that both $\phi_i$ and ρ are in $A^\alpha_q(H^1)$ for all $\alpha > 0$ with $\alpha = \frac{1}{q} - \frac{1}{2}$ . Our proof is based on the assumption that the $\phi_i$ possess an asymptotic smoothness property at the electron-nuclear cusps.