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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Solving PDEs in domains with complex evolving morp
hology: Rothschild Visiting Fellow Lecture - Elli
ott\, C (University of Warwick)
DTSTART;TZID=Europe/London:20150914T160000
DTEND;TZID=Europe/London:20150914T170000
UID:TALK60684AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/60684
DESCRIPTION:Many physical models give rise to the need to solv
e partial differential equations in time dependent
regions. The complex morphology of biological me
mbranes and cells coupled with biophysical mathema
tical models present significant computational cha
llenges as evidenced within the Newton Institute p
rogramme "Coupling Geometric PDEs with Physics for
Cell Morphology\, Motility and Pattern Formation"
. In this talk we discuss the mathematical issues
associated with the formulation of PDEs in time d
ependent domains in both flat and curved space. He
re we are thinking of problems posed on time depen
dent d-dimensional hypersurfaces Gamma(t) in R^{d+
1}. The surface Gamma(t) may be the boundary of t
he bounded open bulk region Omega(t). In this sett
ing we may also view Omega(t) as (d+1)-dimensional
sub-manifold in R^{d+2}. Using this observation w
e may develop a discretisation theory applicable t
o both surface and bulk equations. We will presen
t an abstract framework for treating the theory of
well- posedness of solutions to abstract paraboli
c partial differential equations on evolving Hilbe
rt spaces using generalised Bochner spaces. This
theory is applicable to variational formulations o
f PDEs on evolving spatial domains including movin
g hyper-surfaces. We formulate an appropriate time
derivative on evolving spaces called the material
derivative and define a weak material derivative
in analogy with the usual time derivative in fixed
domain problems\; our setting is abstract and not
restricted to evolving domains or surfaces. Then
we show well-posedness to a certain class of parab
olic PDEs under some assumptions on the parabolic
operator and the data. Specifically\, we study in
turn a surface heat equation\, an equation posed o
n a bulk domain\, a novel coupled bulk-surface sys
tem and an equation with a dynamic boundary condit
ion. We give some background to applications in ce
ll biology. We describe how the theory may be used
in the development and numerical analysis of evol
ving surface finite element spaces which unifies t
he discrtetisation methodology for evolving surfac
e and bulk equations. In order to have good discre
tisation one needs good meshes. We will indicate h
ow geometric PDEs may be used to compute high qual
ity meshes. We give some computational examples
from cell biology involving the coupling of surfa
ce evolution to processes on the surface.\n
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:
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