Un of Tenn | Math Department

Seminars and Colloquiums
for the week of November 30, 2009

Speaker:

L. J. Gray, Computer Science and Mathematics Division, ORNL , Monday
Professor Jamie Sethian, U. California – Berkeley, Friday - cancelled
Zhen-Qing Chen, The University of Washington, Friday

Monday, November 30

Applied Math Seminar
TIME: 3:35 – 4:25 pm
ROOM: HBB 112
SPEAKER: L. J. Gray, Computer Science and Mathematics Division, ORNL
TITLE: “Level Set-Boundary Integral Simulation of the Rayleigh-Taylor Instability”
ABSTRACT: This work is a collaboration with Prof. Maria Garzon (Department of Mathematics, University of Oviedo, Spain) and Prof. James Sethian, Department of Mathematics, University of California, Berkeley CA

Moving boundary problems are an important class of applications for which boundary integral equation methods have inherent advantages. Compared to constructing a volume discretization at every time step,it is a simpler task to re-mesh just the evolving boundary. Moreover, these type of simulations generally require complete knowledge of the

function derivatives (e.g., potential gradient) on the boundary and, as discussed in this talk, these derivatives can be obtained accurately and efficiently from the integral equation formulation.

The Level Set method is powerful technique for tracking surface evolution, a notable feature being the ability to follow topological changes of the domain. Moreover, the Level Set algorithm generates, (mostly) automatically, the boundary integral mesh for the new surface. The coupling of integral equation and Level Set methods is therefore attractive, and this approach has been investigated in several recent articles. This talk will present the results of applying these methods to simulate the Rayleigh-Taylor instability: specifically, the pinch-off of a fluid column based upon an inviscid

potential flow model with capillary forces. The fluid interface velocity is obtained from a linear element Galerkin approximation of the 3D axi-symmetric Laplace equation. As noted above, the post-processing evaluation of the surface gradient is of primary

interest, and the algorithm employed is based upon a boundary limit equation. This approach is fast, avoiding a costly complete boundary integration, and the accuracy is aided by the fact that the singular integrals that do remain do not involve elliptic integrals, and can therefore be evaluated analytically.

The Raleigh-Taylor simulations presented in this talk follow the motion through the first pinch off of the fluid and the subsequent cascade of droplet formation. The scaling behavior near pinch-off is known theoretically, and the calculated critical exponents show excellent agreement with theory.

Finally, it is hoped that results from current work on modeling the recent two-fluid experiments of Burton and Taborek (Phys. Rev. Lett. 101 2008) can be presented.

Friday, December 4

Special Lecture -- CANCELLED
TIME: 1:25 – 2:15 pm
ROOM: HBB 103
SPEAKER: Professor Jamie Sethian, U. California – Berkeley
TITLE: TBA
ABSTRACT: TBA

TIME: 3:35 pm
ROOM: HBB 112
SPEAKER: Zhen-Qing Chen, The University of Washington
TITLE: “Symmetric Markov Processes and Heat Kernel Estimates”
ABSTRACT: In this talk, I will describe recent development of the DeGiorgi-Nash-Moser-Aronson type theory for a class of symmetric discontinuous processes (or equivalently, a class of symmetric integro-differential operators). A prototype of the Markov processes under consideration is the mixture of symmetric diffusion of uniformly elliptic divergence form operator and symmetric stable-like processes on $R^d$. I will focus on the sharp two-sided estimates for the transition density functions (or heat kernels) of the processes, a priori Holder estimate and parabolic Harnack inequalities for their parabolic functions. To establish these results, we employ methods from both probability theory and analysis. Based on joint work with Takashi Kumagai.

If you are interested in giving or arranging a talk for one of our seminars or colloquiums, please review our calendar.

If you have questions, or a date you would like to confirm, please contact Dr. Steve Wise.

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