Past

The probabilistic approach to homogenization can be adapted to fully

degenerate situations, where irreducibility is insured from a Doeblin type

condition. Using recent results on weak sense Poisson equations in a

similar framework, obtained jointly with A. Veretennikov, together with a

regularization procedure, we prove the homogenization result. A similar

approach can also handle degenerate random homogenization.

It is known for elliptic problems with smooth coefficients

that the solution is smooth in the interior of the domain;

low regularity is only possible near the boundary.

The $hp$-version of the FEM allows us to exploit this

property if we use meshes where the element size grows

porportionally to the element's distance to the boundary

and the approximation order is suitably linked to the

element size. In this way most degrees of freedom are

concentrated near the boundary.

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In this talk, we will discuss convergence and complexity

issues of the boundary concentrated FEM. We will show

that it is comparable to the classical boundary element

method (BEM) in that it leads to the same convergence rate

(error versus degrees of freedom). Additionally, it

generalizes the classical FEM since it does not require

explicit knowledge of the fundamental solution so that

it is also applicable to problems with (smooth) variable

coefficients.

Consider a spectrally one-sided Levy process X and reflect it at

its past infimum I. Call this process Y. We determine the law of the

first crossing time of Y of a positive level a in terms of its

'scale' functions. Next we study the exponential decay of the

transition probabilities of Y killed upon leaving [0,a]. Restricting

ourselves to the case where X has absolutely continuous transition

probabilities, we also find the quasi-stationary distribution of

this killed process. We construct then the process Y confined in

[0,a] and prove some properties of this process.

A reveiw of results about spectral analysis of generators of

some stochastic lattice models (a stochastic planar rotators model, a

stochastic Blume-Capel model etc.) will be presented. Then I'll discuss new

results by R.A. Minlos, Yu.G. Kondratiev and E.A. Zhizhina concerning spectral

analysis of the generator of stochastic continuous particle system. The

construction of one-particle subspaces of the generators and the spectral

analysis of the generator restricted on these subspaces will be the focus of

the talk.

Following the framework of Formaggia and Perotto (Numer.

Math. 2001 and 2003), anisotropic a posteriori error estimates have been

proposed for various elliptic and parabolic problems. The error in the

energy norm is bounded above by an error indicator involving the matrix

of the error gradient, the constant being independent of the mesh aspect

ratio. The matrix of the error gradient is approached using

Zienkiewicz-Zhu error estimator. Numerical experiments show that the

error indicator is sharp. An adaptive finite element algorithm which

aims at producing successive triangulations with high aspect ratio is

proposed. Numerical results will be presented on various problems such

as diffusion-convection, Stokes problem, dendritic growth.

Recently Friesecke, James and Muller established the following

quantitative version of the rigidity of SO(n) the group of special orthogonal

matrices. Let U be a bounded Lipschitz domain. Then there exists a constant

C(U) such that for any mapping v in the L2-Sobelev space the L^2-distance of

the gradient controlls the distance of v a a single roation.

This interesting inequality is fundamental in several problems concerning

dimension reduction in nonlinear elasticity.

In this talk, we will present a joint work with Muller and Zhong where we

investigate an analagous quantitative estimate where we replace SO(n) by an

arbitrary smooth, compact and SO(n) invariant subset of the conformal

matrices E. The main novelty is that exact solutions to the differential

inclusion Df(x) in E a.e.x in U are not necessarily affine mappings.

http://www.maths.ox.ac.uk/notices/events/abstracts/stochastic-analysis/…

Stability is central to the study of numerical algorithms for solving
partial differential equations. But stability can be subtle and elusive. In
fact, for a number of important classes of PDE problems, no one has yet
succeeded in devising stable numerical methods. In developing our
understanding of stability and instability, a wide range of mathematical
ideas--with origins as diverse as functional analysis,differential geometry,
and algebraic topology--have been enlisted and developed. The talk will
explore the concept of stability of discretizations to PDE, its significance,
and recent advances in its understanding.

While the average settling velocity of particles in a suspension has been successfully predicted, we are still unsuccessful with the r.m.s velocity, with theories suggesting a divergence with the size of

the container and experiments finding no such dependence. A possible resolution involves stratification originating from the spreading of the front between the clear liquid above and the suspension below. One theory describes the spreading front by a nonlinear diffusion equation

$\frac{\partial \phi}{\partial t} = D \frac{\partial }{\partial z}(\phi^{4/5}(\frac{\partial \phi}{\partial z})^{2/5})$.

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Experiments and computer simulations find differently.