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SUMMARY:Solving PDEs in domains with complex evolving morphology:  Rothsch
 ild Visiting Fellow Lecture - Elliott\, C (University of Warwick)
DTSTART:20150914T150000Z
DTEND:20150914T160000Z
UID:TALK60684@talks.cam.ac.uk
CONTACT:42080
DESCRIPTION:Many physical models give rise to the need to solve partial di
 fferential equations in time dependent regions.  The complex morphology of
  biological membranes and cells coupled with biophysical mathematical mode
 ls present significant computational challenges as evidenced within the Ne
 wton Institute programme "Coupling Geometric PDEs with Physics for Cell Mo
 rphology\, Motility and Pattern Formation".  In this talk we discuss the m
 athematical issues associated with the formulation of PDEs in time depende
 nt domains in both flat and curved space. Here we are thinking of problems
  posed on time dependent d-dimensional hypersurfaces Gamma(t) in R^{d+1}. 
  The surface Gamma(t) may be the boundary of the bounded open bulk region 
 Omega(t). In this setting we may also view Omega(t) as (d+1)-dimensional s
 ub-manifold in R^{d+2}. Using this observation we may develop a discretisa
 tion theory applicable to both surface and bulk equations.  We will presen
 t an abstract framework for treating the theory of well- posedness of solu
 tions to abstract parabolic partial differential equations on evolving Hil
 bert spaces using generalised Bochner spaces.  This theory is applicable t
 o variational formulations of PDEs on evolving spatial domains including m
 oving hyper-surfaces. We formulate an appropriate time derivative on evolv
 ing spaces called the material derivative and define a weak material deriv
 ative in analogy with the usual time derivative in fixed domain problems\;
  our setting is abstract and not restricted to evolving domains or surface
 s. Then we show well-posedness to a certain class of parabolic PDEs under 
 some assumptions on the parabolic operator and the data. Specifically\, we
  study in turn a surface heat equation\, an equation posed on a bulk domai
 n\, a novel coupled bulk-surface system and an equation with a dynamic bou
 ndary condition. We give some background to applications in cell biology. 
 We describe how the theory may be used in the development and numerical an
 alysis of evolving surface finite element spaces which unifies the discrte
 tisation methodology for evolving surface and bulk equations. In order to 
 have good discretisation one needs good meshes. We will indicate how geome
 tric PDEs may be used to compute high quality  meshes. We  give some compu
 tational examples from cell biology  involving the coupling of surface evo
 lution to processes on the surface.\n
LOCATION:Seminar Room 1\, Newton Institute
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