Mathematical Models and Methods in Applied Sciences

We study a problem set in a finely mixed periodic medium, modelling electrical conduction in biological tissues. The unknown electric potential solves standard elliptic equations set in different conductive regions (the intracellular and extracellular spaces), separated by a dielectric surface (the cell membranes), which exhibits both a capacitive and a nonlinear conductive behaviour. Accordingly, dynamical conditions prevail on the membranes, so that the dependence of the solution on the time variable t is not only of parametric character.

As the spatial period of the medium goes to zero, the electric potential approaches in a suitable sense a homogenization limit u₀, which keeps the prescribed boundary data, and solves the equation . This is an elliptic equation containing a term depending on the history of the gradient of u₀; the matrices B⁰, A¹ in it depend on the microstructure of the medium. More exactly, we have that, in the limit, the current is still divergence-free, but it depends on the history of the potential gradient, so that memory effects explicitly appear. The limiting equation also contains a term ℱ keeping trace of the initial data.

The paper presents a posteriori error estimators for the stationary Stokes problem. We consider anisotropic finite element discretizations (i.e. elements with very large aspect ratio) where conventional, isotropic error estimators fail.

Our analysis covers two- and three-dimensional domains, conforming and non-conforming discretizations as well as different elements. This large variety of settings requires different approaches and results in different estimators. Furthermore many examples of finite element pairs that are covered by the analysis are presented.

Lower and upper error bounds form the main result with minimal assumptions on the elements. The lower error bound is uniform with respect to the mesh anisotropy with the exception of nonconforming 3D discretizations made of pentahedra or hexahedra. The upper error bound depends on a proper alignment of the anisotropy of the mesh which is a common feature of anisotropic error estimation.

In the special case of isotropic meshes, the results simplify, and upper and lower error bounds hold unconditionally. Some of the corresponding results seem to be novel (in particular for 3D domains), and cover element pairs of practical importance.

The numerical experiments confirm the theoretical predictions and show the usefulness of the anisotropic error estimators.

A multiscale model and numerical method for computing microstructures with large and inhomogeneous deformation is established, in which the microscopic and macroscopic information is recovered by coupling the finite order rank-one convex envelope and the finite element method. The method is capable of computing microstructures which are locally finite order laminates. Numerical experiments on a double-well problem show that plenty of stress free large deformations can be achieved by microstructures consisting of piecewise simple twin laminates.

We investigate a system of partial differential equations modeling ambipolar plasmas. The ambipolar — or zero current — model is obtained from general plasmas equations in the limit of vanishing Debye length. In this model, the electric field is expressed as a linear combination of macroscopic variable gradients. We establish that the governing equations can be written as a symmetric form by using entropic variables. The corresponding dissipation matrices satisfy the null space invariant property and the system of partial differential equations can be written as a normal form, i.e. in the form of a symmetric hyperbolic–parabolic composite system. By properly modifying the chemistry source terms and/or the diffusion matrices, asymptotic stability of equilibrium states is established and decay estimates are obtained. We also establish the continuous dependence of global solutions with respect to vanishing electron mass.

In the earlier paper (Part I) we study the homogenization of the Mullins–Sekerka problem for deterministic particle distributions in the case that the effective range of particle interactions is smaller than the system size. Here we extend those results to the case of stochastic particle distributions. The main difficulty is that in this case particles collide with positive probability. We show, however, that colliding particles are few and do not influence the macroscopic behavior.