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 "elbatnuocnu" backwards in the box below (case sensitive):

Archive

Show me a random blog post
 2020 

Jul 2020

Happy √3ϕ+π-e Approximation Day!

May 2020

A surprising fact about quadrilaterals
Interesting tautologies

Mar 2020

Log-scaled axes

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

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

Archive

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