We study a non-local version of the Cahn-Hilliard dynamics for phase separation in a two-component incompressible and immiscible mixture with linear mobilities. In difference to the celebrated local model with nonlinear mobility, it is only assumed that the divergences of the two fluxes -but not necessarily the fluxes themselves -annihilate each other. Our main result is a rigorous proof of existence of weak solutions. The starting point is the formal representation of the dynamics as a constrained gradient flow in the Wasserstein metric. We then show that time-discrete approximations by means of the incremental minimizing movement scheme converge to a weak solution in the limit. Further, we compare the non-local model to the classical Cahn-Hilliard model in numerical experiments. Our results illustrate the significant speed-up in the decay of the free energy due to the higher degree of freedom for the velocity fields.∇ · J tot = 0 where J tot = J 1 + J 2 1991 Mathematics Subject Classification. 35K65, 35K41, 49J40, 76T99.

This work is devoted to the numerical study of the Cahn-Hilliard equation with dynamic boundary conditions. A spatial finite-volume discretization is proposed which couples a 2d-method in a smooth connected domain and a 1d-method on its boundary. The convergence of the sequence of approximate solutions is proved and various numerical simulations are given.

Abstract. "Discrete Duality Finite Volume" schemes (DDFV for short) on general 2D meshes, in particular non conforming ones, are studied for the Stokes problem with Dirichlet boundary conditions. The DDFV method belongs to the class of staggered schemes since the components of the velocity and the pressure are approximated on different meshes. In this paper, we investigate from a numerical and theoretical point of view, whether or not the stability condition holds in this framework for various kind of mesh families. We obtain that different behaviors may occur depending on the geometry of the meshes.For instance, for conforming acute triangle meshes, we prove the unconditional Inf-Sup stability of the scheme, whereas for some conforming or non-conforming Cartesian meshes we prove that Inf-Sup stability holds up to a single unstable pressure mode. In any cases, the DDFV method appears to be very robust.

We propose a finite element scheme for numerical approximation of degenerate parabolic problems in the form of a nonlinear anisotropic Fokker-Planck equation. The scheme is energy-stable, only involves physically motivated quantities in its definition, and is able to handle general unstructured grids. Its convergence is rigorously proven thanks to compactness arguments, under very general assumptions. Although the scheme is based on Lagrange finite elements of degree 1, it is locally conservative after a local postprocess giving rise to an equilibrated flux. This also allows to derive a guaranteed a posteriori error estimate for the approximate solution. Numerical experiments are presented in order to give evidence of a very good behavior of the proposed scheme in various situations involving strong anisotropy and drift terms.