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 accurate and efficient hybrid central-upwind (HCU) type schemes for the numerical resolution of a two-fluid model commonly used by the nuclear and petroleum industry. Particularly, we propose a method which does not make use of any information about the eigenstructure 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 evolution equation which describes how pressure evolves in time. By applying a quasi-staggered Lax-Friedrichs type discretization for this pressure equation together with a Modified Lax-Friedrich type discretization of the convective terms, we obtain a central type scheme which allows to cope with the nonlinearity (nonlinear pressure waves) of the two-fluid model in a robust manner. Then, in order to obtain an accurate resolution of mass fronts, we employ a modification of the convective mass fluxes by hybridizing the central type mass flux components with upwind type components. This hybridization is based on a splitting of the mass fluxes into components corresponding to the pressure and volume fraction variables, recovering an accurate resolution of a contact discontinuity. In the numerical simulations, the resulting HCU scheme gives results comparable to an approximate Riemann solver while being superior in efficiency. Furthermore, the HCU scheme yields better robustness 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 equation in $\mathbb{R}^d$, d=2 or 3, using backward Euler's scheme. For this discretization, we derive a residual indicator, which use a spatial residual indicator based on the jumps of normal and tangential derivatives of the nonconforming approximation 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 the exact controllability of a finite dimensional system obtained by discretizing in space and time the linear 1-D wave system with a boundary control at one extreme. It is known that usual schemes obtained with finite difference or finite element methods are not uniformly controllable with respect to the discretization parameters h and Δt. We introduce an implicit finite difference scheme which differs from the usual centered one by additional terms of order h2 and Δt2. Using a discrete version of Ingham's inequality for nonharmonic Fourier series and spectral properties of the scheme, we show that the associated control can be chosen uniformly bounded in L2(0,T) and in such a way that it converges to the HUM control of the continuous wave, i.e. the minimal L2-norm control. The results are illustrated with 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.