Mathematical Models and Methods in Applied Sciences

We prove convergence of a computable adaptive wavelet algorithm for the solution of elliptic PDE's, which combines Richardson type iterations with nonlinear projection steps.

This paper analyzes two algorithms based on the S₂ approximation for the evolutive, gray, 1D radiative transfer equations in the slab with isotropic scattering. It is known that simple fixed point iteration does not work when the medium is optically thick, and that this difficulty can be overcome by using a Newton method. In this paper we quantify the first observation, giving a criterium to decide a priori whether fixed point converges or not. We also observe that the Newton method can be implemented faster than the fixed point iteration, but this implementation strongly relies on the particular discretization schemes used here. Existence theorems for the discrete nonlinear problem are also proved.

This paper deals with the qualitative analysis of a model related to the immune response to the evolution of the progression of endothelial cells which have lost their differentiation and start their evolution toward methastatic states. We prove the existence of solutions to the Cauchy problem related to the model. The asymptotic behavior in time of our solutions is also investigated.

A nonlinear stability analysis of thermal convection in a saturated porous medium is presented. Density is assumed to have a cubic temperature dependence and the equations of flow in the porous medium are described via Darcy's law with a Forchheimer drag term. Unconditional stability is established using L³ and L⁴ norms and it is shown that L² theory is insufficient to obtain similar results. Previous authors have established conditional nonlinear stability but we believe this is the first analysis that addresses the important problem of unconditional stability for the system in hand.

We study here a model of conservative nonlinear conservation law with a flux function with discontinuous coefficients, namely the equation u_t + (k(x)u(1 - u))_x = 0. It is a particular entropy condition on the line of discontinuity of the coefficient k which ensures the uniqueness of the entropy solution. This condition is discussed and justified. On the other hand, we perform a numerical analysis of the problem. Two finite volume schemes, the Godunov scheme and the VFRoe-ncv scheme, are proposed to simulate the conservation law. They are compared with two finite volume methods classically used in an industrial context. Several tests confirm the good behavior of both new schemes, especially through the discontinuity of permeability k (whereas a loss of accuracy may be detected when industrial methods are performed). Moreover, a modified MUSCL method which accounts for stationary states is introduced.

In this paper, the existence of solution for a coupled system of variational inequalities with highly oscillating coefficients modelling a micro-elastohydrodynamic lubrication problem is stated. Moreover, by means of two-scale convergence techniques, the limit coupled models when the frequency of oscillation tends to zero are established for different amplitude-frequency rates. The effect of roughness appears to be limited to one of the coupled nonlinear equations, either the elastic one or the hydrodynamic one. Finally, some numerical examples to illustrate both the qualitative behavior of the solution and the theoretical convergence results are presented.