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PDEs on evolving domains

Presented by: 
Charlie Elliott
Wednesday 22nd July 2015 - 09:00 to 10:15
INI Seminar Room 1
Recently with Aphonse and Stinner we have presented an abstract framework for treating the theory of well- posedness of solutions to abstract parabolic partial differential equations on evolving Hilbert spaces. This theory is applicable to variational for- mulations of PDEs on evolving spatial domains including moving 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 set- ting is abstract and not restricted to evolving domains or surfaces. 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 domain, a novel coupled bulk-surface system and an equation with a dynamic boundary condition. In this talk we give some background to applications in cell biology. We describe how the theory may be used in the development and numerical analysis of evolving surface finite element methods and give some computational examples involving the coupling of surface evolution to processes on the surface. We indicate how this approach may work for PDEs on general evolving domains. We will discuss briefly surface finite elements for finding surfaces in the context of models for biomembranes. We will indicate how other approaches might be applicable.

Background material: K. P. Deckelnick, G. Dziuk and C. M. Elliott Computation of Geometric PDEs and Mean Curvature Flow Acta Numerica (2005) 139–232
G. Dziuk and C. M. Elliott Finite element methods for surface partial differential equations Acta Numerica (2013) 289–396

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons