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Ranner, Thomas (2013) Computational surface partial differential equations. PhD thesis, University of Warwick.
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Official URL: http://webcat.warwick.ac.uk/record=b2689212~S1
Abstract
Surface partial differential equations model several natural phenomena; for
example in
uid mechanics, cell biology and material science. The domain of the
equations can often have complex and changing morphology. This implies analytic
techniques are unavailable, hence numerical methods are required. The aim of this
thesis is to design and analyse three methods for solving different problems with
surface partial differential equations at their core.
First, we define a new finite element method for numerically approximating
solutions of partial differential equations in a bulk region coupled to surface partial
differential equations posed on the boundary of this domain. The key idea is to take
a polyhedral approximation of the bulk region consisting of a union of simplices,
and to use piecewise polynomial boundary faces as an approximation of the surface
and solve using isoparametric finite element spaces. We study this method in the
context of a model elliptic problem. The main result in this chapter is an optimal
order error estimate which is confirmed in numerical experiments.
Second, we use the evolving surface finite element method to solve a Cahn-
Hilliard equation on an evolving surface with prescribed velocity. We start by deriving
the equation using a conservation law and appropriate transport formulae and
provide the necessary functional analytic setting. The finite element method relies
on evolving an initial triangulation by moving the nodes according to the prescribed
velocity. We go on to show a rigorous well-posedness result for the continuous equations
by showing convergence, along a subsequence, of the finite element scheme.
We conclude the chapter by deriving error estimates and present various numerical
examples.
Finally, we stray from surface finite element method to consider new unfitted
finite element methods for surface partial differential equations. The idea is to use a
fixed bulk triangulation and approximate the surface using a discrete approximation
of the distance function. We describe and analyse two methods using a sharp interface
and narrow band approximation of the surface for a Poisson equation. Error
estimates are described and numerical computations indicate very good convergence
and stability properties.
Item Type: | Thesis (PhD) | ||||
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Subjects: | Q Science > QA Mathematics Q Science > QC Physics |
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Library of Congress Subject Headings (LCSH): | Differential equations, Partial, Finite element method, Surfaces | ||||
Official Date: | July 2013 | ||||
Dates: |
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Institution: | University of Warwick | ||||
Theses Department: | Mathematics Institute | ||||
Thesis Type: | PhD | ||||
Publication Status: | Unpublished | ||||
Supervisor(s)/Advisor: | Elliott, Charles M. | ||||
Sponsors: | Engineering and Physical Sciences Research Council (EPSRC) | ||||
Extent: | xi,191 leaves : illustrations | ||||
Language: | eng |
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