for one and two dimensional elliptic interface problems based on Cartesian
triangulations. The key is to modify the basis functions so that the
homogeneous jump conditions are satisfied in the presence of discontinuity
in the coefficients. Both non-conforming and conforming
finite element spaces are considered. Corresponding interpolation
functions are proved to be second order accurate in the maximum norm.
For non-homogeneous jump conditions, we have developed a new strategy to
transform the original interface problem to a new one with homogeneous jump
conditions using the level set function.
If time permits, I will also explain some recent progress in this direction
including the augmented IFEM for piecewise constant coefficient, and a SVD
free version of the method.
cup of tea to the internal dynamics of the earth. In this talk, I
will discuss a few experiments where boundaries to the fluid play
surprising roles in changing the behaviors of a classical Rayleigh-
Bénard convection system. In one, mobile boundaries lead to
regular large-scale oscillations that involve the entire system.
This could be related to the continental kinetics on earth over
the past two billion years, as super-continents formed and
broke apart in cyclic fashion. In another experiment, we found that
seemingly impeding partitions in thermal convection can boost the
overall heat transport by several folds, once the partitions are
properly arranged, thanks to an unexpected symmetry-breaking
suspension of particles down an incline, which is described by a system
of conservation laws equipped with an equilibrium theory for particle
settling and resuspension. Singular shock appears in the high particle
concentration case that relates to the particle-rich ridge observed in
the experiments. We analyze the formation of the singular shock as well
as its local structure, and extend to the finite volume case, which
leads to a linear relationship between the shock front with time to the
one-third power. We then add the surface tension effect into the model
and show how it regularizes the singular shock via numerical
movement is one of the most prevailing observations in nature. Yet, despite
considerable progress, many of the theoretical principles underlying the
emergence of large scale synchronization among moving individuals are still
poorly understood. For example, a key question in the study of animal motion is
how the details of locomotion, interaction between individuals and the environment
contribute to the macroscopic dynamics of the hoard, flock or swarm. The
talk will present some of the prevailing models for swarming and collective
motion with emphasis on stochastic descriptions. The goal is to identify some generic characteristics
regarding the build-up and maintenance of collective order in swarms. In
particular, whether order and disorder correspond to different phases,
requiring external environmental changes to induce a transition, or rather meta-stable
states of the dynamics, suggesting that the emergence of order is kinetic.
Different aspects of the phenomenon will be presented, from experiments with locusts
to our own attempts towards a statistical physics of collective motion.