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- Giving to Math

Event Type:

Applied Mathematics and Computation Seminar

Date/Time:

Monday, November 2, 2009 - 04:00 to 05:00

Location:

GILK 113

Guest Speaker:

Institution:

University of Nevada, Reno

Abstract:

In this talk I will briefly introduce our group, its motivation

and goals, some results that we obtained so far, and what we hope

to accomplish in the future.

In particular, I will explain why we believe that it makes sense

to design numerical methods that are independent of the partial

differential equations (PDE) solved. The current standard in my

field is to develop sophisticated error estimates and techniques

that only work (or work efficiently) for a narrow class of PDE.

In contrast to that, our methods work in the same way for any

PDE or multiphysics PDE system, and they can even be used in

other fields such as computer graphics or robotics. Examples

will be shown.

Main ideas of adaptive higher-order finite element methods

(hp-FEM) will be explained and their extremely fast, exponential

convergence will be demonstrated via live computations and

compared to the slow, algebraic convergence of traditional

low-order FEM.

I will mention the main ideas of our novel adaptive multimesh

hp-FEM that makes it possible to discretize a PDE system monolithically

(i.e., as if it was a single equation) while using individual

(optimal) meshes for all physical fields in the system. Numerical

comparisons to standard (single-mesh) hp-FEM will be presented.

We will show that even fluid flows can be handled as multiphysics

problems with velocity components and pressure approximated using

different meshes.

We will show that the multimesh hp-FEM makes it possible to design

simple and efficient adaptive algorithms for time-dependent PDE

problems based on dynamically-changing meshes. (Automatic adaptivity

for time-dependent problems is much more difficult than adaptivity

for statonary ones because of the necessity of mesh unrefinement).

I will explain the main idea and show several computational videos

related to various problems (flame propagation, thermally-conductive

flow, microwave heating, thermoelasticity, single and two-phase flow,

heat and moisture transfer in concrete, Gross-Pitaevski equation

of quantum physics, etc).

If some time is left, I will say a few words about our open source

project FEMhub whose objective is to create an open source distribution

of finite element codes with unified Python interface.