Danielle Hilhorst

Finite dimensional exponential attractor for the phase field model

We consider the phase field equations in arbitrary space dimension. We show that the corresponding boundary value problems are well-posed when assuming that the initial data is square integrable and prove the existence of a maximal attractor and of an inertial set.

Spatial segregation limit of a competition–diffusion system

We consider a competition–diffusion system and study its singular limit as the interspecific competition rate tend to infinity. We prove the convergence to a Stefan problem with zero latent heat.

The finite volume method for Richards equation

In this paper we prove the convergence of a finite volume scheme for the discretization of an elliptic–parabolic problem, namely Richards equation β(P)t−div(K(β(P))× ∇(P+z))=0, together with Dirichlet boundary conditions and an initial condition. This is done by means of a priori estimates in L2 and the use of Kolmogorovs theorem on relative compactness of subsets of L2.

A combined finite volume–nonconforming/mixed-hybrid finite element scheme for degenerate parabolic problems

We propose and analyze a numerical scheme for nonlinear degenerate parabolic convection–diffusion–reaction equations in two or three space dimensions. We discretize the diffusion term, which generally involves an inhomogeneous and anisotropic diffusion tensor, over an unstructured simplicial mesh of the space domain by means of the piecewise linear nonconforming (Crouzeix–Raviart) finite element method, or using the stiffness matrix of the hybridization of the lowest-order Raviart–Thomas mixed finite element method. The other terms are discretized by means of a cell-centered finite volume scheme on a dual mesh, where the dual volumes are constructed around the sides of the original mesh. Checking the local Péclet number, we set up the exact necessary amount of upstream weighting to avoid spurious oscillations in the convection-dominated case. This technique also ensures the validity of the discrete maximum principle under some conditions on the mesh and the diffusion tensor. We prove the convergence of the scheme, only supposing the shape regularity condition for the original mesh. We use a priori estimates and the Kolmogorov relative compactness theorem for this purpose. The proposed scheme is robust, only 5-point (7-point in space dimension three), locally conservative, efficient, and stable, which is confirmed by numerical experiments.

The Nishiura--Ohnishi Free Boundary Problem in the 1D Case

A free boundary problem due to Nishiura and Ohnishi is solved in one space dimension. That problem was derived, during their study of phase separation phenomena in diblock copolymers, as an asymptotic limit of pattern-forming PDEs generalizing that of Cahn and Hilliard. The free boundary problem in one dimension reduces to a linear system of ODEs for the lengths of the intervals between interfaces. This system also arises in a completely different context as the spatial discretization of a simple heat equation in a medium with periodic properties. (The medium is homogeneous in an important special case.) The initial-value problem for this system is completely solved, and global stability results for stationary solutions (in which the interfaces are regularly spaced) are obtained. Nucleation phenomena are briefly discussed.

A competition-diffusion system approximation to the classical two-phase Stefan problem

A new type of competition-diffusion system with a small parameter is proposed. By singular limit analysis, it is shown that any solution of this system converges to the weak solution of the two-phase Stefan problem with reaction terms. This result exhibits the relation between an ecological population model and water-ice solidification problems.

A reaction–diffusion system with fast reversible reaction

Abstract We consider a reaction–diffusion system which models a fast reversible reaction between two mobile reactants and prove convergence of the solutions as the reaction rate tends to infinity, where the limiting problem is given by a diffusion equation with nonlinear diffusion. Since the rate function has no sign, the usual methods to obtain a priori estimates in the case of irreversible reactions do not apply; we deduce instead a priori estimates from computations based on Lyapunov function techniques.

Vanishing latent heat limit in a Stefan-like problem arising in biology

We consider a two phase Stefan problem with a reaction term in arbitrary space dimension and prove that as the latent heat coefficient tends to zero, its weak solution converges to the weak solution of the corresponding problem with zero latent heat, which is obtained as the spatial segregation limit of a competition-diffusion system. In particular, we obtain a uniform convergence result for the corresponding interfaces in the one-dimensional case.

On interacting populations that disperse to avoid crowding: preservation of segregation

Abstract : This document considers a mathematical model for interacting biological species that disperse as a response to population pressure. The authors demonstrate an interesting feature of the model: species which are initially segregated remain segregated for all time.

Mass conserving Allen–Cahn equation and volume preserving mean curvature flow

We consider a mass conserved Allen-Cahn equation