MaPSS: Maths Postgraduate Seminar Series: Eric Hester -- Understanding multi-scale Partial Differential Equations on arbitrary domains using the straightforward differential geometry of the signed distance function

Hello all, 

We have our first MaPSS seminar of the semester at 17:00 on Monday the 4th March in
Carslaw 535.  I’ll be presenting some of my work from the ANZIAM conference this year,
and I promise it’ll be a great opportunity to see an interesting talk, meet some
fellow postgrads, and get some free pizza! 

************************************************************************************** 

Speaker 1: Eric Hester 

Title: Understanding multi-scale Partial Differential Equations on arbitrary domains
using the straightforward differential geometry of the signed distance function.  

Abstract: Partial Differential Equations (PDEs) are a core part of mathematical
modelling in science, industry and engineering; the applications are endless! And often
the most interesting problems involve complex geometries.  We normally model them using
PDEs which live on separate domains, with boundary conditions applied at the
infinitesimal interfaces.  

But that’s math, not reality.  Reality is smooth! Things get fuzzy at the micro and
nanoscale.  And it can also be useful to do simulations this way -- don’t try to
simulate your PDEs on complicated domains.  Instead, perturb your PDEs to *implicitly*
model your boundary conditions.  

Having smooth (but small) transitions between domains means we are considering
inherently multi-scale singular perturbations of PDEs.  So these approximations are only
true asymptotically.  What we need to know is how these smooth approximations behave in
the limit.  Do they tend to the correct answer? How fast? And do they work in arbitrary
geometries? 

This talk will examine a really useful coordinate system for analysing such multi-scale
PDEs.  It all comes from the straightforward differential geometry of the signed
distance function.  I’ll be focussing on examples from my research on modelling moving
objects in fluid dynamics.  No background in differential geometry or fluid dynamics is
required.  

************************************************************************************** 

Hope to see you there! 

Details can also be found on the school’s new Postgraduate Society website:
http://www.maths.usyd.edu.au/u/MaPS/mapss.html 

Cheers, Eric