mscroggs.co.uk
mscroggs.co.uk

subscribe

Blog

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

Archive

Show me a random blog post
 2020 

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

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

Archive

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