mscroggs.co.uk
mscroggs.co.uk

subscribe

Blog

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:
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.
Previous post in series
PhD thesis, chapter 5
This is the sixth post in a series of posts about my PhD thesis.

Similar posts

PhD thesis, chapter 5
PhD thesis, chapter 4
PhD thesis, chapter 3
PhD thesis, chapter 2

Comments

Comments in green were written by me. Comments in blue were not written by me.
 Add a Comment 


I will only use your email address to reply to your comment (if a reply is needed).

Allowed HTML tags: <br> <a> <small> <b> <i> <s> <sup> <sub> <u> <spoiler> <ul> <ol> <li>
To prove you are not a spam bot, please type "v" then "e" then "c" then "t" then "o" then "r" in the box below (case sensitive):

Archive

Show me a random blog post
 2020 

Feb 2020

PhD thesis, chapter ∞
PhD thesis, chapter 5
PhD thesis, chapter 4
PhD thesis, chapter 3
Inverting a matrix
PhD thesis, chapter 2

Jan 2020

PhD thesis, chapter 1
Gaussian elimination
Matrix multiplication
Christmas (2019) is over
 2019 
▼ show ▼
 2018 
▼ show ▼
 2017 
▼ show ▼
 2016 
▼ show ▼
 2015 
▼ show ▼
 2014 
▼ show ▼
 2013 
▼ show ▼
 2012 
▼ show ▼

Tags

binary interpolation folding paper news trigonometry probability braiding matt parker matrices manchester science festival matrix of minors fractals plastic ratio arithmetic pizza cutting menace error bars dates triangles wave scattering rhombicuboctahedron cross stitch talking maths in public european cup geometry oeis computational complexity map projections the aperiodical noughts and crosses sorting books speed tmip inverse matrices advent calendar chalkdust magazine christmas bubble bobble electromagnetic field light determinants php signorini conditions bempp mathsteroids realhats boundary element methods ucl reuleaux polygons radio 4 hannah fry flexagons javascript cambridge golden ratio harriss spiral game show probability mathsjam preconditioning pythagoras approximation coins stickers machine learning video games games go manchester london mathslogicbot gerry anderson weak imposition polynomials palindromes asteroids martin gardner rugby finite element method chebyshev raspberry pi folding tube maps reddit chess platonic solids hexapawn misleading statistics phd numerical analysis bodmas inline code national lottery data visualisation hats statistics football world cup gaussian elimination data matrix of cofactors twitter royal institution sport dataset draughts latex frobel curvature christmas card accuracy graph theory simultaneous equations golden spiral logic sound estimation london underground python people maths a gamut of games dragon curves puzzles big internet math-off royal baby game of life tennis propositional calculus sobolev spaces craft matrix multiplication weather station final fantasy countdown programming nine men's morris wool captain scarlet ternary pac-man

Archive

Show me a random blog post
▼ show ▼
© Matthew Scroggs 2012–2020