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.

In this paper we present a methodology for constructing accurateand efficient hybrid central-upwind (HCU) type schemes forthe numerical resolution of a two-fluid model commonly used by thenuclear and petroleum industry. Particularly, we propose a methodwhich does not make use of any information about theeigenstructure of the Jacobian matrix of the model. The two-fluid model possesses a highly nonlinear pressure law. From the mass conservation equations we develop an evolutionequation which describes how pressure evolves in time. By applyinga quasi-staggered Lax-Friedrichs type discretization for thispressure equation together with a Modified Lax-Friedrich typediscretization of the convective terms, we obtain a central typescheme which allows to cope with the nonlinearity (nonlinearpressure waves) of the two-fluid model in a robust manner.Then, in order to obtain an accurate resolution of mass fronts, weemploy a modification of the convective mass fluxes by hybridizingthe central type mass flux components with upwind type components.This hybridization is based on a splitting of the mass fluxes intocomponents corresponding to the pressure and volume fractionvariables, recovering an accurate resolution of a contactdiscontinuity. In the numerical simulations, the resulting HCU scheme givesresults comparable to an approximate Riemann solver while beingsuperior in efficiency. Furthermore, the HCU scheme yields betterrobustness than other popular Riemann-free upwind schemes.

We address here mathematical models related to the Laser-Plasma Interaction. After a simplified introduction to the physical background concerning the modelling of the laser propagation and its interaction with a plasma, we recall some classical results about the geometrical optics in plasmas. Then we deal with the well known paraxial approximation of the solution of the Maxwell equation; we state a coupling model between the plasma hydrodynamics and the laser propagation. Lastly, we consider the coupling with the ion acoustic waves which has to be taken into account to model the so called Brillouin instability. Here, besides the macroscopic density and the velocity of the plasma, one has to handle the space-time envelope of the main laser wave, the space-time envelope of the stimulated Brillouin backscattered laser wave and the space envelope of the Brillouin ion acoustic waves. Numerical methods are also described to deal with the paraxial model and the three-wave coupling system related to the Brillouin instability.

The paper presents an a posteriori error estimator for a (piecewise linear) nonconforming finite element approximation of the heat equationin $\mathbb{R}^d$ , d=2 or 3,using backward Euler's scheme. For this discretization, we derive a residual indicator, which usea spatial residual indicator based on thejumps of normal and tangential derivatives of the nonconformingapproximation and a time residual indicator based on the jump of broken gradients at each time step.Lower and upper bounds form the main results with minimal assumptions on the mesh.Numerical experiments and a space-time adaptive algorithm confirm the theoretical predictions.

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.

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.

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.