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
This is the sixth post in a series of posts about my PhD thesis.
                        
(Click on one of these icons to react to this blog post)

You might also enjoy...

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> <logo>
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
 2025 

Jan 2025

Christmas (2024) is over
Friendly squares
 2024 
▼ show ▼
 2023 
▼ show ▼
 2022 
▼ show ▼
 2021 
▼ show ▼
 2020 
▼ show ▼
 2019 
▼ show ▼
 2018 
▼ show ▼
 2017 
▼ show ▼
 2016 
▼ show ▼
 2015 
▼ show ▼
 2014 
▼ show ▼
 2013 
▼ show ▼
 2012 
▼ show ▼

Tags

finite element method databet gather town exponential growth manchester php turtles datasaurus dozen the aperiodical asteroids mathsteroids mathsjam advent calendar dates football map projections matrix of minors python machine learning inline code final fantasy go triangles statistics runge's phenomenon raspberry pi radio 4 golden spiral harriss spiral phd recursion realhats latex graphs craft manchester science festival data anscombe's quartet pizza cutting chebyshev folding paper frobel numerical analysis matt parker inverse matrices probability numbers nine men's morris menace preconditioning palindromes binary a gamut of games bubble bobble london underground speed hexapawn countdown data visualisation royal baby simultaneous equations dragon curves logo pi friendly squares logic errors quadrilaterals wave scattering wool bodmas misleading statistics cross stitch guest posts rhombicuboctahedron mathslogicbot fonts bots newcastle matrices puzzles flexagons coins polynomials trigonometry martin gardner javascript convergence ucl pythagoras oeis ternary christmas card matrix of cofactors crochet golden ratio fence posts books geogebra braiding mean games talking maths in public gerry anderson interpolation draughts rugby chalkdust magazine finite group stickers matrix multiplication gaussian elimination crossnumber sound dinosaurs zines national lottery video games game of life bempp pac-man reuleaux polygons regular expressions folding tube maps correlation pascal's triangle 24 hour maths graph theory squares chess european cup reddit curvature captain scarlet royal institution cambridge people maths sobolev spaces signorini conditions plastic ratio electromagnetic field christmas live stream weak imposition edinburgh error bars programming stirling numbers kings approximation big internet math-off hyperbolic surfaces determinants logs game show probability propositional calculus sorting standard deviation london tmip sport boundary element methods accuracy arithmetic platonic solids hats weather station noughts and crosses youtube pi approximation day tennis light geometry world cup computational complexity hannah fry dataset fractals estimation news

Archive

Show me a random blog post
▼ show ▼
© Matthew Scroggs 2012–2025