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 "j" then "u" then "m" then "p" in the box below (case sensitive):

Archive

Show me a random blog post
 2021 

Jan 2021

Christmas (2020) is over
 2020 
▼ show ▼
 2019 
▼ show ▼
 2018 
▼ show ▼
 2017 
▼ show ▼
 2016 
▼ show ▼
 2015 
▼ show ▼
 2014 
▼ show ▼
 2013 
▼ show ▼
 2012 
▼ show ▼

Tags

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

Archive

Show me a random blog post
▼ show ▼
© Matthew Scroggs 2012–2021