The finite element approximation of optimal control problems forsemilinear elliptic partial differential equation is considered,where the control belongs to a finite-dimensional set and stateconstraints are given in finitely many points of the domain. Underthe standard linear independency condition on the active gradientsand a strong second-order sufficient optimality condition, optimalerror estimates are derived for locally optimal controls.

Consider the domain $Z_\epsilon=\{x\in\mathbb{R}^n ; {dist}(x,\epsilon\mathbb{Z}^n)> \epsilon^\gamma\}$ and let the free path length be defined as $\tau_\epsilon(x,v)=\inf\{t> 0 ; x-tv\in Z_\epsilon\}.$ In the Boltzmann-Grad scaling corresponding to $\gamma=\frac{n}{n-1}$ , it is shownthat the limiting distribution $\phi_\epsilon$ of $\tau_\epsilon$ is bounded from belowby an expression of the form C/t, for some C> 0. A numerical study seems toindicate that asymptotically for large t, $\phi_\epsilon\sim C/t$ .This is an extension of a previous work [J. Bourgain et al., Comm. Math. Phys.190 (1998) 491-508]. As aconsequence, it is proved that the linear Boltzmann type transport equation is inappropriate to describethe Boltzmann-Grad limit of the periodic Lorentz gas, at variance with the usualcase of a Poisson distribution of scatterers treated in [G. Gallavotti (1972)].

This paper deals with the diffusion limit of a kinetic equation where thecollisions are modeled by a Lorentz type operator. The main aim is to construct adiscrete scheme to approximate this equation which gives for any value of theKnudsen number, and in particular at the diffusive limit, the right discretediffusion equation with the same value of the diffusion coefficient as in thecontinuous case. We are also naturally interested with a discretization whichcan be used with few velocity discretization points, in order to reduce the cost ofcomputation.

Saccular aneurisms, swelling of a blood vessel,are investigated in order (i) to estimate the development risk ofthe wall lesion, before and after intravascular treatment,assuming that the pressure is the major factor,and (ii) to better plan medical interventions.Numerical simulations, using the finite element method,are performed in three-dimensional aneurisms.Computational meshes are derived from medical imaging datato take into account both between-subject and within-subjectanatomical variability of the diseased vessel segment.The 3D reconstruction is associated with a faceted surface.A geometrical model is then obtained to be finally meshed for a finite element use. The pulsatile flow of incompressible Newtonian blood is illustrated by numerical simulations carried out in two saccular aneurism types, a side- and a terminal-aneurism.High pressure zones are observed in the aneurism cavity,especially in the terminal one.

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.

This contribution deals with a mixed variational formulation of 3D contact problems with the simplest model involving friction. This formulation is based on a dualization of the set of admissible displacements and the regularization of the non-differentiable term. Displacements are approximated by piecewise linear elements while the respective dual variables by piecewise constant functions on a dual partition of the contact zone. The rate of convergence is established provided that the solution is smooth enough. The numerical realization of such problems will be discussed and results of a model example will be shown.


A new finite element, which is continuously differentiable, but only piecewise quadratic polynomials on a type of uniform triangulations, is introduced. We construct a local basis which does not involve nodal values nor derivatives. Different from the traditional finite elements, we have to construct a special, averaging operator which is stable and preserves quadratic polynomials. We show the optimal order of approximation of the finite element in interpolation, and in solving the biharmonic equation. Numerical results are provided confirming the analysis.


We consider the magnetic induction equation for the evolution of a magnetic field in a plasma where the velocity is given. The aim is to design a numerical scheme which also handles the divergence constraint in a suitable manner. We design and analyze an upwind scheme based on the symmetrized version of the equations in the non-conservative form. The scheme is shown to converge to a weak solution of the equations. Furthermore, the discrete divergence produced by the scheme is shown to be bounded. We report several numerical experiments that show that the stable upwind scheme of this paper is robust.

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.

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.

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 present a hybrid finite-volume-particle numerical method for computing the transport of a passive pollutant by a flow. The flow is modeled by the one- and two-dimensional Saint-Venant system of shallow water equations and the pollutant propagation is described by a transport equation.This paper is an extension of our previous work [Chertock, Kurganov and Petrova, J. Sci. Comput. (to appear)], where the one-dimensional finite-volume-particle method has been proposed.The core idea behind the finite-volume-particle method is to use different schemes for the flow and pollution computations: the shallow water equations are numerically integrated using a finite-volume scheme, while the transport equation is solved by a particle method. This way the specific advantages of each scheme are utilizedat the right place. A special attention is given to the recovery of the point values of the numerical solution from its particle distribution. The reconstruction is obtained using a dual equation for the pollutant concentration.This results in a significantly enhanced resolution of the computed solution and also makes it much easier to extendthe finite-volume-particle method to the two-dimensional case.

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 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.

The integration to steady state of many initial value ODEs and PDEs using the forward Euler methodcan alternatively be considered as gradient descent for an associated minimization problem.Greedy algorithms such as steepest descent for determining the step size are asslow to reach steady state as is forward Euler integration with the best uniform step size.But other, much faster methods using bolder step size selection exist.Various alternatives are investigated from both theoretical and practical points of view.The steepest descent method is also known for the regularizing or smoothing effect that thefirst few steps have for certain inverse problems,amounting to a finite time regularization. We further investigate the retention of thisproperty using the faster gradient descent variants in the context of two applications.When the combination of regularization and accuracy demands more than a dozen or so steepestdescent steps, the alternatives offer an advantage, even though (indeed because)the absolute stability limit of forward Euler is carefully yet severely violated.

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 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 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.

In this paper, the convergence of a Neumann-Dirichlet algorithm to approximateCoulomb's contact problem between two elastic bodies is proved in a continuous setting. In this algorithm, the natural interface between the two bodies is retained as a decomposition zone.

In this paper, we propose a new numerical method for solving elliptic equations in unbounded regions of ${\mathbb{R}}^n$. The method is based on the mapping of a part of the domain into a bounded region. An appropriate family of weighted spaces is used for describing the growth or the decay of functions at large distances. After exposing the main ideas of the method, we analyse carefully its convergence. Some 3D computational results are displayed to demonstrate its efficiency and its high performance.