In this paper we investigate the motion of a rigid ball in an incompressible perfect fluid occupying ${\mathbb R}^2$. We prove the global in time existence and the uniqueness of the classical solution for this fluid-structure problem. The proof relies mainly on weighted estimates for the vorticity associated with the strong solution of a fluid-structure problem obtained by incorporating some dissipation.

In this paper, we are concerned with a kind of Signorini transmission problem in a unbounded domain. A variational inequality is derived when discretizing this problem by coupled FEM-BEM. To solve such variational inequality, an iterative method, which can be viewed as a variant of the D-N alternative method, will be introduced. In the iterative method, the finite element part and the boundary element part can be solved independently. It will be shown that the convergence speed of this iteration is independent of the mesh size. Besides, a combination between this method and the steepest descent method is also discussed.

The aim of this paper is to present a method using both the ideas of sectionalapproach and moment methods in order to accurately simulate evaporationphenomena in gas-droplets flows. Using the underlying kinetic interpretation ofthe sectional method [Y. Tambour, Combust. Flame60 (1985)15–28] exposed in [F. Laurent and M. Massot, Combust. Theory Model.5 (2001) 537–572], we propose an extension of thisapproach based on a more accurate representation of the droplet size numberdensity in each section ensuring the exact conservation of two moments (asopposed to only one moment used in the classical approach). A correspondingsecond-order numerical scheme, with respect to space and droplet size variables,is also introduced and can be proved to be positive and to satisfy a maximumprinciple on the velocity and the mean droplet mass under a suitable CFL-likecondition. Numerical simulations have been performed and the results confirm theaccuracy of this new method even when a very coarse mesh for the droplet sizevariable (i.e.: a low number of sections) is used.

The purpose of this work is to study an example of low Mach (Froude) number limit of compressible flows when the initial density (height) is almost equal to a function depending on x. This allows us to connect the viscous shallow water equation and the viscous lake equations. More precisely, we study this asymptotic with well prepared data in a periodic domain looking at the influence of the variability of the depth. The result concerns weak solutions. In a second part, we discuss the general low Mach number limit for standard compressible flows given in P.–L. Lions' book that means with constant viscosity coefficients.

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.

In this paper, we are interested in the modelling and the finite element approximation of a petroleum reservoir, in axisymmetric form. The flow in the porous medium is governed by the Darcy-Forchheimer equation coupled with a rather exhaustive energy equation. The semi-discretized problem is put under a mixed variational formulation, whose approximation is achieved by means of conservative Raviart-Thomas elements for the fluxes and of piecewise constant elements for the pressure and the temperature. The discrete problem thus obtained is well-posed and a posteriori error estimates are also established. Numerical tests are presented validating the developed code.

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.

We present the convergence analysis of locally divergence-free discontinuous Galerkin methodsfor the induction equations which appear in the ideal magnetohydrodynamic system. When we use a second order Runge Kutta time discretization, under the CFL condition $\Delta t\sim h^{4/3}$ , we obtain error estimates in L 2 of order $\mathcal{O} (\Delta t^2 + h^{m + 1/2})$ where m is the degree of the local polynomials.

We discuss the occurrence of oscillationswhen using central schemes of the Lax-Friedrichs type (LFt), Rusanov's method and the staggered andnon-staggered second order Nessyahu-Tadmor (NT) schemes. Although these schemes are monotone or TVD, respectively, oscillations may be introduced at local data extrema.The dependence of oscillatory properties on the numerical viscositycoefficient is investigated rigorously for the LFt schemes, illuminating alsothe properties of Rusanov's method. It turns out, that schemes with a large viscosity coefficient areprone to oscillations at data extrema. For all LFt schemes except for the classicalLax-Friedrichs method, occurring oscillations are damped in the course of a computation.This damping effect also holds for Rusanov's method. Concerning the NT schemes, the non-staggeredversion may yield oscillatory results, while it can be shown rigorously that the staggered NTscheme does not produce oscillations when using the classical minmod-limiter under arestriction on the time step size. Note that this restriction is not thesame as the condition ensuring the TVD property. Numerical investigations of one-dimensional scalar problems and of the system of shallow water equations in two dimensions with respect to the phenomenon complete the paper.

We propose a Diphasic Low Mach Number (DLMN) system for the modelling of diphasic flows without phase change at low Mach number, system which is an extension of the system proposed by Majda in [Center of Pure and Applied Mathematics, Berkeley, report No. 112] and [Combust. Sci. Tech.42 (1985) 185–205] for low Mach number combustion problems. This system is written for a priori any equations of state. Under minimal thermodynamic hypothesis which are satisfied by a large class of generalized van der Waals equations of state, we recover some natural properties related to the dilation and to the compression of bubbles. We also propose an entropic numerical scheme in Lagrangian coordinates when the geometry is monodimensional and when the two fluids are perfect gases. At last, we numerically show that the DLMN system may become ill-posed when the entropy of one of the two fluids is not a convex function.

This paper studies theexact controllability of a finite dimensional system obtained bydiscretizing in space and time the linear 1-D wave system with aboundary control at one extreme. It is known that usual schemesobtained with finite difference or finite element methods are notuniformly controllable with respect to the discretizationparameters h and Δt. We introduce an implicit finitedifference scheme which differs from the usual centered one byadditional terms of order h 2 and Δt 2. Using a discreteversion of Ingham's inequality for nonharmonic Fourier series andspectral properties of the scheme, we show that the associatedcontrol can be chosen uniformly bounded in L2(0,T) and in sucha way that it converges to the HUM control of the continuous wave,i.e. the minimal L 2-norm control. The results are illustratedwith several numerical experiments.

Preconditioners for hyperbolic systems are numerical artifacts to accelerate the convergence to a steady state.In addition, the preconditioner should also be included in the artificial viscosity or upwinding terms to improve the accuracy of the steady state solution. For time dependent problemswe use a dual time stepping approach. The preconditioner affects the convergence rate and the accuracy of the subiterations within each physical time step. We considertwo types of local preconditioners:Jacobi and low speed preconditioning.We can express the algorithm in several sets of variableswhile using only the conservation variables for the flux terms.We compare the effect of these various variable setson the efficiency and accuracy of the scheme.

Finite element approximation for degenerate parabolic equations is considered. We propose a semidiscrete scheme provided with order-preserving and L1 contraction properties, making use of piecewise linear trial functions and the lumping mass technique. Those properties allow us to apply nonlinear semigroup theory, and the wellposedness and stability in L1 and L∞, respectively, of the scheme are established. Under certain hypotheses on the data, we also derive L1 convergence without any convergence rate. The validity of theoretical results is confirmed by numerical examples.

The magnetization of a ferromagnetic sample solves anon-convex variational problem, where its relaxation by convexifyingthe energy density resolves relevantmacroscopic information. The numerical analysis of the relaxed modelhas to deal with a constrained convexbut degenerated, nonlocal energy functional in mixed formulation formagnetic potential u and magnetization m.In [C. Carstensen and A. Prohl, Numer. Math.90(2001) 65–99], the conforming P1 - (P0)d -element in d=2,3 spatialdimensions is shown to lead toan ill-posed discrete problem in relaxed micromagnetism, and suboptimalconvergence.This observation motivated anon-conforming finite element method which leads toa well-posed discrete problem, with solutions converging atoptimal rate.In this work, we provide both an a priori and a posteriori error analysis for twostabilized conforming methods which account for inter-element jumps of thepiecewise constant magnetization.Both methods converge at optimal rate;the new approach is applied to a macroscopic nonstationary ferromagnetic model [M. Kružík and A. Prohl, Adv. Math. Sci. Appl. 14 (2004) 665–681 – M. Kružík and T. Roubíček, Z. Angew. Math. Phys. 55 (2004) 159–182 ].

On exhibe dans cette note une paramétrix (au sens faible) de l'opérateur sous-jacent à l'équation CFIE de l'électromagnétisme. L'intérêt de cetteparamétrix est de se prêter à différentes stratégies de discrétisationet ainsi de pouvoir être utilisée comme préconditionneur de la CFIE.On montre aussi que l'opérateur sous-jacent à la CFIE satisfait une conditionInf-Sup discrète uniforme, applicable aux espaces de discrétisation usuellement rencontrésen électromagnétisme, et qui permet d'établir un résultat inédit de convergencenumérique de la CFIE.

We present a finite volume method based on the integration of the Laplaceequation on both the cells of a primal almost arbitrary two-dimensionalmesh and those of adual mesh obtained by joining the centers of the cells of the primal mesh.The key ingredient is the definition of discrete gradient and divergenceoperators verifying a discrete Green formula.This method generalizes an existing finite volume method thatrequires “Voronoi-type” meshes.We show the equivalence of this finite volume method with a non-conformingfinite element method with basis functions being P 1 on the cells,generally called “diamond-cells”, of a third mesh. Under geometrical conditions on these diamond-cells, we prove a first-order convergence both in the $\xHone_0$ normand in the L² norm. Superconvergence results are obtained on certain types of homothetically refined grids. Finally, numerical experiments confirm these results and also show second-order convergencein the L² norm on general grids.They also indicate that this method performs particularly well for the approximationof the gradient of the solution, and may be used on degenerating triangular grids.An example of application on non-conforming locally refined grids is given.

We build corotational symmetric solutions to the harmonic map flow from the unit disc into the unit sphere which have constant degree. First, we prove the existence of such solutions, using a time semi-discrete scheme based on the idea that the harmonic map flow is the L2-gradient of the relaxed Dirichlet energy. We prove a partial uniqueness result concerning these solutions. Then, we compute numerically these solutions by a moving-mesh method which allows us to deal with the singularity at the origin. We show numerical evidence of the convergence of the method.

We propose a general approach for the numerical approximation ofoptimal control problems governed by a linear advection–diffusionequation, based on a stabilization method applied to theLagrangian functional, rather than stabilizing the state andadjoint equations separately. This approach yields a coherentlystabilized control problem. Besides, it allows a straightforward a posteriori error estimate in which estimates of higher order termsare needless. Our a posteriori estimates stems from splitting theerror on the cost functional into the sum of an iteration errorplus a discretization error. Once the former is reduced below agiven threshold (and therefore the computed solution is “near”the optimal solution), the adaptive strategy is operated on thediscretization error. To prove the effectiveness of the proposedmethods, we report some numerical tests, referring to problems inwhich the control term is the source term of theadvection–diffusion equation.

In this paper, the Babuška's theory of Lagrange multipliers is extended to higher order elliptic Dirichlet problems. The resulting variational formulation provides an efficient numerical squeme in meshless methods for the approximation of elliptic problems with essential boundary conditions.

In this paper, we extend the reduced-basis methods and associated a posteriori error estimators developed earlier for elliptic partial differential equations to parabolic problems with affine parameter dependence. The essential new ingredient is the presence of time in the formulation and solution of the problem – we shall “simply” treat time as an additional, albeit special, parameter. First, we introduce the reduced-basis recipe – Galerkin projection onto a space WN spanned by solutions of the governing partial differential equation at N selected points in parameter-time space – and develop a new greedy adaptive procedure to “optimally” construct the parameter-time sample set. Second, we propose error estimation and adjoint procedures that provide rigorous and sharp bounds for the error in specific outputs of interest: the estimates serve a priori to construct our samples, and a posteriori to confirm fidelity. Third, based on the assumption of affine parameter dependence, we develop offline-online computational procedures: in the offline stage, we generate the reduced-basis space; in the online stage, given a new parameter value, we calculate the reduced-basis output and associated error bound. The operation count for the online stage depends only on N (typically small) and the parametric complexity of the problem; the method is thus ideally suited for repeated, rapid, reliable evaluation of input-output relationships in the many-query or real-time contexts.