Past

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.