We design efficient numerical schemes for approximating the MHD equations in multi-dimensions. Numerical approximations must be able to deal with the complex wave structure of the MHD equations and the divergence constraint. We propose schemes based on the genuinely multi-dimensional (GMD) framework of [S. Mishra and E. Tadmor, Commun. Comput. Phys. 9 (2010) 688–710; S. Mishra and E. Tadmor, SIAM J. Numer. Anal. 49 (2011) 1023–1045]. The schemes are formulated in terms of vertex-centered potentials. A suitable choice of the potential results in GMD schemes that preserve a discrete version of divergence. First- and second-order divergence preserving GMD schemes are tested on a series of benchmark numerical experiments. They demonstrate the computational efficiency and robustness of the GMD schemes.

Les méthodes sans maillage emploient une interpolation associée à un ensemble de particules : aucune information concernant la connectivité ne doit être fournie.Un des atouts de ces méthodes est que la discrétisationpeut être enrichie d'une façon très simple, soit en augmentant le nombre de particules (analogue à lastratégie de raffinement h), soit en augmentant l'ordre de consistance (analogueà la stratégie de raffinement p). Néanmoins, le coût du calcul des fonctionsd'interpolation est très élevé et ceci représente un inconvénient vis-à-visdes éléments finis. Cet article présente une interpolation mixte élémentsfinis-particules qui résulte de la généralisation de plusieurs travaux dans cedomaine. La formulation de cette interpolation mixte est valable pour n'importe quel ordre de consistance. Dans ce contexte, onénonce un estimateur d'erreur a priori dont la démonstrationse base dans les propriétés de l'interpolation mixte.Ce résultat permet d'étudier la convergence de laméthode d'enrichissement et d'établir les stratégies de raffinement del'interpolation qui permettent d'atteindre une solutionavec une précision satisfaisante.

We present in this paper a proof of well-posedness and convergence for the parallelSchwarz Waveform Relaxation Algorithm adapted to an N-dimensional semilinear heatequation. Since the equation we study is an evolution one, each subproblem at each stephas its own local existence time, we then determine a common existence time for everyproblem in any subdomain at any step. We also introduce a new technique: Exponential DecayError Estimates, to prove the convergence of the Schwarz Methods, with multisubdomains,and then apply it to our problem.

The aim of this paper is to derive a general model for reduced viscous and resistive Magnetohydrodynamics (MHD) and to study its mathematical structure. The model is established for arbitrary density profiles in the poloidal section of the toroidal geometry of Tokamaks. The existence of global weak solutions, on the one hand, and the stability of the fundamental mode around initial data, on the other hand, are investigated.

We prove a posteriori error estimates of optimal order for linearSchrödinger-type equations in the L ∞(L 2)- and theL ∞(H 1)-norm. We discretize only in time by theCrank-Nicolson method. The direct use of the reconstructiontechnique, as it has been proposed by Akrivis et al. in [Math. Comput.75 (2006) 511–531], leads to a posteriori upper bounds thatare of optimal order in the L ∞(L 2)-norm, but ofsuboptimal order in the L ∞(H 1)-norm. The optimality inthe case of L ∞(H 1)-norm is recovered by using anauxiliary initial- and boundary-value problem.

We present a Gause type predator–prey model incorporating delay due to response of prey population growth to density and gestation. The functional response of predator is assumed to be of Holling type II. In absence of prey, predator has a density dependent death rate. Sufficient criterion for uniform persistence is derived. Conditions are found out for which system undergoes a Hopf–bifurcation.

We consider the flow of a viscous incompressible fluid through a rigidhomogeneous porous medium. The permeability of the medium dependson the pressure, so that the model is nonlinear. We propose a finiteelement discretization of this problem and, in the case where thedependence on the pressure is bounded from above and below, we proveits convergence to the solution and propose an algorithm to solvethe discrete system. In the case where the dependenceon the pressure is exponential, we propose a splittingscheme which involves solving two linear systems, but parts of the analysis of this method are still heuristic. Numerical tests are presented, which illustrate the introduced methods.


Caused by the problem of unilateral contact during vibrations of satellite solar arrays, the aim of this paper is to better understand such a phenomenon. Therefore, it is studied here a simplified model composed by a beam moving between rigid obstacles. Our purpose is to describe and compare some families of fully discretized approximations and their properties, in the case of non-penetration Signorini's conditions. For this, starting from the works of Dumont and Paoli, we adapt to our beam model the singular dynamic method introduced by Renard. A particular emphasis is given in the use of a restitution coefficient in the impact law. Finally, various numerical results are presented and energy conservation capabilities of the schemes are investigated.

In this paper we prove the discrete compactness property for a discontinuous Galerkin approximation of Maxwell's systemon quite general tetrahedral meshes. As a consequence, a discrete Friedrichs inequality is obtainedand the convergence of the discrete eigenvalues to the continuous ones is deducedusing the theory of collectively compact operators.Some numerical experiments confirm the theoretical predictions.

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.

"Least regret control" consists in trying tofind a control which "optimizes the situation" with the constraint of not making things tooworse with respect to a known reference control,in presence of more or less significantperturbations. This notion was introduced in [7].It is recalled on a simple example (an ellipticsystem, with distributed control and boundary perturbation) in Section 2. We show that the problem reduces to a standard optimal control problem for augmented state equations.On another hand, we have introduced in recentnotes [9-12] the method ofvirtual control, aimed at the"decomposition of everything" (decomposition ofthe domain, of the operator, etc). Anintroduction to this method is presented, withouta priori knowledge needed, in Sections 3 and 4,directly on the augmented state equations.For problems without control, or with "standard" control, numerical applications of the virtualcontrol ideas have been given in the notes[9-12] and in the note[5].One of the first systematic paper devoted to allkind of decomposition methods, including multicriteria, is a jointpaper with A. Bensoussan and R. Temam, towhom this paper is dedicated, cf. [1].

The powerful Hamilton-Jacobi theory is used for constructing regularizations and error estimates for optimal design problems. The constructed Pontryagin method is a simple and general method for optimal design and reconstruction: the first, analytical, step is to regularize the Hamiltonian; next the solution to its stationary Hamiltonian system, a nonlinear partial differential equation, is computed with the Newton method. The method is efficient for designs where the Hamiltonian function can be explicitly formulated and when the Jacobian is sparse, but becomes impractical otherwise (e.g. for non local control constraints). An error estimate for the difference between exact and approximate objective functions is derived, depending only on the difference of the Hamiltonian and its finite dimensional regularization along the solution path and its L2 projection, i.e. not on the difference of the exact and approximate solutions to the Hamiltonian systems.

In this paper we construct a new H(div)-conforming projection-based p-interpolation operator that assumes only Hr(K) $\cap$${\bf \tilde H}$-1/2(div, K)-regularity (r > 0) on the reference element (either triangle or square) K. We show that this operator is stable with respect to polynomial degrees and satisfies the commuting diagram property. We also establish an estimate for the interpolation error in the norm of the space ${\bf \tilde H}$-1/2(div, K), which is closely related to the energy spaces for boundary integral formulations of time-harmonic problems of electromagnetics in three dimensions.

We derive asymptotic formulas for the solutions of the mixed boundary value problem forthe Poisson equation on the union of a thin cylindrical plate and several thin cylindricalrods. One of the ends of each rod is set into a hole in the plate and the other one issupplied with the Dirichlet condition. The Neumann conditions are imposed on the wholeremaining part of the boundary. Elements of the junction are assumed to have contrastingproperties so that the small parameter, i.e. the relative thickness,appears in the differential equation, too, while the asymptotic structures cruciallydepend on the contrastness ratio. Asymptotic error estimates are derived in anisotropicweighted Sobolev norms.

We propose and analyze numerical schemes for viscosity solutions of time-dependent Hamilton-Jacobi equations on the Heisenberg group.The main idea is to construct a grid compatible with the noncommutative group geometry. Under suitable assumptions on the data, the Hamiltonian and the parameters for the discrete first order scheme,we prove that the error between the viscosity solution computed at the grid nodes and the solution of the discrete problem behaves like $\sqrt{h}$ where h is the mesh step. Such an estimate is similar to those available in the Euclidean geometrical setting.The theoretical results are tested numerically on some examples for which semi-analytical formulas for the computation of geodesics are known. Other simulations are presented, for both steady and unsteady problems.

This paper deals with the numerical solution of nonlinear Black-Scholes equation modeling European vanilla call option pricing under transaction costs. Using an explicit finite difference scheme consistent with the partial differential equation valuation problem, a sufficient condition for the stability of the solution is given in terms of the stepsize discretization variables and the parameter measuring the transaction costs. This stability condition is linked to some properties of the numerical approximation of the Gamma of the option, previously obtained. Results are illustrated with numerical examples.

In this paper, we derive and analyze a Reissner-Mindlin-like model for isotropic heterogeneous linearly elastic plates. The modeling procedure is based on a Hellinger-Reissner principle, which we modify to derive consistent models. Due to the material heterogeneity, the classical polynomial profiles for the plate shear stress are replaced by more sophisticated choices, that are asymptotically correct. In the homogeneous case we recover a Reissner-Mindlin model with 5/6 as shear correction factor. Asymptotic expansions are used to estimate the modeling error. We remark that our derivation is not based on asymptotic arguments only. Thus, the model obtained is more sophisticated (and accurate) than simply taking the asymptotic limit of the three dimensional problem. Moreover, we do not assume periodicity of the heterogeneities.

This paper is concerned with the numerical approximation of mean curvature flow t → Ω(t) satisfying an additional inclusion-exclusion constraint Ω1 ⊂ Ω(t) ⊂ Ω2. Classical phase field model to approximate these evolving interfaces consists in solving the Allen-Cahn equation with Dirichlet boundary conditions. In this work, we introduce a new phase field model, which can be viewed as an Allen Cahn equation with a penalized double well potential. We first justify this method by a Γ-convergence result and then show some numerical comparisons of these two different models.

We first prove an abstract result for a class of nonlocalproblems using fixed point method. We apply this result toequations revelant from plasma physic problems. These equationscontain terms like monotone or relative rearrangement of functions.So, we start the approximation study by using finite element todiscretize this nonstandard quantities. We end the paper by giving a numerical resolution of a model containing those terms.

This work deals with a system of nonlinear parabolic equations arisingin turbulence modelling. The unknowns are the N components of the velocityfield u coupled with two scalar quantities θ and φ. The systempresents nonlinear turbulent viscosity $A(\theta,\varphi)$ and nonlinearsource terms of the form $\theta^2|\nabla u|^2$ and $\theta\varphi|\nabla u|^2$ lying in L 1. Some existence results are shown in this paper, including $L^\infty$ -estimates and positivity for both θ and φ.