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PhD thesis, chapter ∞
2020-02-17
This is the sixth post in a series of posts about my PhD thesis.
Before we move on to some concluding remarks and notes about possible future work,
I must take this opportunity to thank my co-supervisors Timo Betcke and Erik Burman,
as without their help, support, and patience, this work would never have happened.
Future work
There are of course many things related to the work in my thesis that could be worked on in the future by me or others.
In the thesis, we presented the analysis of the weak imposition of Dirichlet, Neumann, mixed Dirichlet–Neumann, Robin, and Signorini boundary conditions on Laplace's equation;
and Dirichlet, and mixed Dirichlet–Neumann conditions on the Helmholtz equation. One area of future work would be to extend this analysis to other conditions, such as the imposition of
Robin conditions on the Helmholtz equation. It would also be of great interest to extend the method to other problems, such as Maxwell's equations. For Maxwell's equations, it looks like the
analysis will be significantly more difficult.
In the problems in later chapters, in particular chapter 4, the ill-conditioning of the matrices obtained from the method led to slow or even inaccurate solutions.
It would be interesting to look into alternative preconditioning methods for these problems as a way to improve the conditioning of these matrices. Developing these preconditioners appears to be
very important for Maxwell's equations: general the matrices involved when solving Maxwell's equations tend to be very badly ill-conditioned, and in the few experiments I ran to try out
the weak imposition of boundary conditions on Maxwell's equations, I was unable to get a good solution due to this.
Your work
If you are a undergraduate or master's student and are interested in working on similar stuff to me, then you could look into
doing a PhD with Timo and/or Erik (my supervisors).
There are also many other people around working on similar stuff, including:
- Iain Smears, David Hewett, and others in the Department of Mathematics at UCL;
- Garth Wells in the Department of Engineering at Cambridge;
- Patrick Farrell, Carolina Urzua-Torres, and others in the Department of Mathematics at Oxford;
- Euan Spence, Ivan Graham, and others in the Department of Mathematics at Bath;
- Stéphanie Chaillat-Loseille, and others at ENSTA in Paris;
- Marie Rognes, and others at Simula in Oslo.
There are of course many, many more people working on this, and this list is in no way exhaustive. But hopefully this list can be a useful starting point for anyone interested in studying this area
of maths.
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