mscroggs.co.uk
mscroggs.co.uk

subscribe

Blog

PhD thesis, chapter 5

 2020-02-16 
This is the fifth post in a series of posts about my PhD thesis.
In the fifth and final chapter of my thesis, we look at how boundary conditions can be weakly imposed on the Helmholtz equation.

Analysis

As in chapter 4, we must adapt the analysis of chapter 3 to apply to Helmholtz problems. The boundary operators for the Helmholtz equation satisfy less strong conditions than the operators for Laplace's equation (for Laplace's equation, the operators satisfy a condition called coercivity; for Helmholtz, the operators satisfy a weaker condition called Gårding's inequality), making proving results about Helmholtz problem harder.
After some work, we are able to prove an a priori error bound (with \(a=\tfrac32\) for the spaces we use):
$$\left\|u-u_h\right\|\leqslant ch^{a}\left\|u\right\|$$

Numerical results

As in the previous chapters, we use Bempp to show that computations with this method match the theory.
The error of our approximate solutions of a Dirichlet (left) and mixed Dirichlet–Neumann problems in the exterior of a sphere with meshes with different values of \(h\). The dashed lines show order \(\tfrac32\) convergence.

Wave scattering

Boundary element methods are often used to solve Helmholtz wave scattering problems. These are problems in which a sound wave is travelling though a medium (eg the air), then hits an object: you want to know what the sound wave that scatters off the object looks like.
If there are multiple objects that the wave is scattering off, the boundary element method formulation can get quite complicated. When using weak imposition, the formulation is simpler: this one advantage of this method.
The following diagram shows a sound wave scattering off a mixure of sound-hard and sound-soft spheres. Sound-hard objects reflect sound well, while sound-soft objects absorb it well.
A sound wave scattering off a mixture of sound-hard (white) and sound-soft (black) spheres.
If you are trying to design something with particular properties—for example, a barrier that absorbs sound—you may want to solve lots of wave scattering problems on an object on some objects with various values taken for their reflective properties. This type of problem is often called an inverse problem.
For this type of problem, weakly imposing boundary conditions has advantages: the discretisation of the Calderón projector can be reused for each problem, and only the terms due to the weakly imposed boundary conditions need to be recalculated. This is an advantages as the boundary condition terms are much less expensive (ie they use much less time and memory) to calculate than the Calderón term that is reused.

This concludes chapter 5, the final chapter of my thesis. Why not celebrate reaching the end by cracking open the following figure before reading the concluding blog post.
An acoustic wave scattering off a sound-hard champagne bottle and a sound-soft cork.
Previous post in series
PhD thesis, chapter 4
This is the fifth post in a series of posts about my PhD thesis.
Next post in series
PhD thesis, chapter ∞

Similar posts

PhD thesis, chapter 4
PhD thesis, chapter 3
PhD thesis, chapter ∞
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 "htdiw" backwards in the box below (case sensitive):

Archive

Show me a random blog post
 2020 

Jul 2020

Happy √3-ϕ+3 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

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

Archive

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