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2020-01-31
 This is the first post in a series of posts about my PhD thesis.
Yesterday, I handed in the final version of my PhD thesis. This is the first in a series of blog posts in which I will attempt to explain what my thesis says with minimal mathematical terminology and notation.
The aim of these posts is to give a more general audience some idea of what my research is about. If you are looking to build on my work, I recommend that you read my thesis, or my papers based on parts of it for the full mathematical detail. You may find these posts helpful with developing a more intuitive understanding of some of the results in these.
In this post, we start at the beginning and look at what is contained in chapter 1 of my thesis.

### Introduction

In general, my work looks at using discretisation to approximately solve partial differential equations (PDEs). We primarily focus on three PDEs:

If $$u$$ represents the temperature in a region, the region contains no heat sources, and the heat distribution in the region is not changing, then $$u$$ a solution of Laplace's equation. Because of this application, Laplace's equation is sometimes called the steady-state heat equation.

The Helmholtz equation models acoustic waves travelling through a medium. The unknown $$u$$ represents the amplitude of the wave at each point. The equation features the wavenumber $$k$$, which gives the number of waves per unit distance.

Maxwell's equations describe the behaviour of electromagnetic waves. The unknown $$e$$ in Maxwell's equatons is an unknown vector, whereas the unknown $$u$$ in Laplace and Helmholtz is a scalar. This adds some difficulty when working with Maxwell's equations.

### Discretisation

In many situations, no method for finding the exact solution to these equations is known. It is therefore very important to be able to get accurate approximations of the solutions of these equations. My PhD thesis focusses on how these can be approximately solved using discretisation.
Each of the PDEs that we want to solve describes a function that has a value at every point in a 3D space: there are an (uncountably) infinite number of points, so these equations effectively have an infinite number of unknown values that we want to know. The aim of discretisation is to reduce the number of unknowns to a finite number in such a way that solving the finite problem gives a good approximation of the true solution. The finite problem can then be solved using your favourite matrix method.
For example, we could split a 2D shape into lots of small triangles, or split a 3D shape into lots of small tetrahedra. We could then try to approximate our solution with a constant inside each triangle or tetrahedron. Or we could approximate by taking a value at each vertex of the triangles or tetrahedra, and a linear function inside shape. Or we could use quadratic functions or higher polynomials to approximate the solution. This is the approach taken by a method called the finite element method.
My thesis, however, focusses on the boundary element method, a method that is closely related to the finite element method. Imagine that we want to solve a PDE problem inside a sphere. For some PDEs (such as the three above), it is possible to represent the solution in terms of some functions on the surface of the sphere. We can then approximate the solution to the problem by approximating these functions on the surface: we then only have to discretise the surface and not the whole of the inside of the sphere. This method will also work for other 3D shapes, and isn't limited to spheres.
One big advantage of this method is that it can also be used if we want to solve a problem in the (infinte) region outside an object, as even if the region is infinite, the surface of the object will be finite. The finite element method has difficulties here, as splitting the infinite region in tetrahedra would require an infinite number of tetrahedra, which destroys the point of discretisation (although this kind of problem can be solved using finite elements if you split into tetrahedra far enough outwards, then use some tricks where your tetrahedron stop). The finite element method, on the other hand, works for a much wider range of PDEs, so in many situations it would be the only of these two methods that can be used.
We focus on the boundary element method as when it can be used, it can be a very powerful method.

### Sobolev spaces

Much of the first chapter of my thesis looks at the definitions of Sobolev function spaces: these spaces describe the types of functions that we will look for to solve our PDEs. It is important to frame the problems using Sobolev spaces, as properties of these spaces are used heavily when proving results about our approximations.
The definitions of these spaces are too technical to include here, but the spaces can be thought of as a way to require certain things of the solutions to our problems. We start be demanding that the function be square integrable: they can be squared the integrated to give a finite answer. Other spaces are then defined by demanding that a function can be differentiated to give a square integrable derivative; or it can be differentiated twice or more times.

### Boundary element method

We now look at how the boundary element method works in more detail. The "boundary" in the name refers to the surface of the object or region that we are solving problems in.
In order to be able to use the boundary element method, we need to know the Green's function of the PDE in question. (This is what limits the number of problems that the boundary element method can be used to solve.) The Green's function is the solution as every point in 3D (or 2D if you prefer) except for one point, where it has an infinite value, but a very special type of infinite value. The Green's function gives the field due to a unit point source at the chosen point.
If a region contains no sources, then the solution inside that region can be represented by adding up point sources at every point on the boundary (or in other words, integrating the Green's function times a weighting function, as the limit of adding up these at lots of points is an integral). The solution could instead be represented by doing the same thing with the derivative of the Green's function.
We define two potential operators—the single layer potential $$\mathcal{V}$$ and the double layer potential $$\mathcal{K}$$—to be the two operators representing the two boundary integrals: these operators can be applied to weight functions on the boundary to give solutions to the PDE in the region. Using these operators, we write a representation formula that tells us how to compute the solution inside the region once we know the weight functions on the boundary.
By looking at the values of these potential operators on the surface, and derivative of these values, (and integrating these over the boundary) we can define four boundary operators: the single layer operator $$\mathsf{V}$$, the double layer operator $$\mathsf{K}$$, the adjoint double layer operator $$\mathsf{K}'$$, the hypersingular operator $$\mathsf{W}$$. We use these operators to write the boundary integral equation that we derive from the representation formula. It is this boundary integral equation that we will apply discretisation to to find an approximation of the weight functions; we can then use the representation formula to find the solution to the PDE.

### The Calderón projector

Using the four boundary operators, we can define the Calderón projector, $$\mathsf{C}$$:
$$\mathsf{C}=\begin{bmatrix} \tfrac12-\mathsf{K}&\mathsf{V}\\ \mathsf{W}&\tfrac12+\mathsf{K}' \end{bmatrix}$$
This is the Calderón projector for problems in a finite interior region. For exterior problems, the exterior Cadleró is used. This is given by
$$\mathsf{C}=\begin{bmatrix} \tfrac12+\mathsf{K}&-\mathsf{V}\\ -\mathsf{W}&\tfrac12-\mathsf{K}' \end{bmatrix}.$$
We treat the Calderón operator like a matrix, even though its entries are operators not numbers. The Calderón projector has some very useful properties that are commonly used when using the boundary element method. These include:
• $$\mathsf{C}^2=\mathsf{C}$$.
• If we apply $$\mathsf{C}$$ to a solution of the PDE, the Calderón projector does nothing to the input.
• If we apply $$\mathsf{C}$$ to any functions, the result will be a solution of the PDE.

These properties conclude chapter 1 of my thesis. Why not take a break and fill the following diagram with hot liquid before reading on. The results of applying the Calderón projector to two functions. By the properties above, these are (the boundary traces of) a solution of the PDE.

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2020-02-04
 This is the second post in a series of posts about my PhD thesis.
During my PhD, I spent a lot of time working on the open source boundary element method Python library Bempp. The second chapter of my thesis looks at this software, and some of the work we did to improve its performance and to make solving problems with it more simple, in more detail.

### Discrete spaces

We begin by looking at the definitions of the discrete function spaces that we will use when performing discretisation. Imagine that the boundary of our region has been split into a mesh of triangles. (The pictures in this post show a flat mesh of triangles, although in reality this mesh will usually be curved.)
We define the discrete spaces by defining a basis function of the space. The discrete space will have one of these basis functions for each triangle, for each edge, or for each vertex (or a combination of these) and the space is defined to contain all the sums of multiples of these basis functions.
The first space we define is DP0 (discontinuous polynomials of degree 0). A basis function of this space has the value 1 inside one triangle, and has the value 0 elsewhere; it looks like this:
Next we define the P1 (continuous polynomials of degree 1) space. A basis function of this space has the value 1 at one vertex in the mesh, 0 at every other vertex, and is linear inside each triangle; it looks like this:
Higher degree polynomial spaces can be defined, but we do not use them here.
For Maxwell's equations, we need different basis functions, as the unknowns are vector functions. The two most commonly spaces are RT (Raviart–Thomas) and NC (Nédélec) spaces. Example basis functions of these spaces look like this: RT (left) and NC (right) basis functions.

### Preconditioning

Suppose we are trying to solve $$\mathbf{A}\mathbf{x}=\mathbf{b}$$, where $$\mathbf{A}$$ is a matrix, $$\mathbf{b}$$ is a (known) vector, and $$\mathbf{x}$$ is the vector we are trying to find. When $$\mathbf{A}$$ is a very large matrix, it is common to only solve this approximately, and many methods are known that can achieve good approximations of the solution. To get a good idea of how quickly these methods will work, we can calculate the condition number of the matrix: the condition number is a value that is big when the matrix will be slow to solve (we call the matrix ill-conditioned); and is small when the matrix will be fast to solve (we call the matrix well-conditioned).
The matrices we get when using the boundary element method are often ill-conditioned. To speed up the solving process, it is common to use preconditioning: instead of solving $$\mathbf{A}\mathbf{x}=\mathbf{b}$$, we can instead pick a matrix $$\mathbf{P}$$ and solve $$\mathbf{P}\mathbf{A}\mathbf{x}=\mathbf{P}\mathbf{b}.$$ If we choose the matrix $$\mathbf{P}$$ carefully, we can obtain a matrix $$\mathbf{P}\mathbf{A}$$ that has a lower condition number than $$\mathbf{A}$$, so this new system could be quicker to solve.
When using the boundary element method, it is common to use properties of the Calderón projector to work out some good preconditioners. For example, the single layer operator $$\mathsf{V}$$ when discretised is often ill-conditioned, but the product of it and the hypersingular operator $$\mathsf{W}\mathsf{V}$$ is often better conditioned. This type of preconditioning is called operator preconditioning or Calderón preconditioning.
If the product $$\mathsf{W}\mathsf{V}$$ is discretised, the result is $$\mathbf{W}\mathbf{M}^{-1}\mathbf{V},$$ where $$\mathbf{W}$$ and $$\mathbf{V}$$ are discretisations of $$\mathsf{W}$$ and $$\mathsf{V}$$, and $$\mathbf{M}$$ is a matrix called the mass matrix that depends on the discretisation spaces used to discretise $$\mathsf{W}$$ and $$\mathsf{V}$$.
In our software Bempp, the mass matrices $$\mathbf{M}$$ are automatically included in product like this, which makes using preconditioning like this easier to program.
As an alternative to operator preconditioning, a method called mass matrix preconditioning is often used: this method uses the inverse mass matrix $$\mathbf{M}^{-1}$$ as a preconditioner (so is like the operator preconditioning example without the $$\mathbf{W}$$).

### More discrete spaces

As the inverse mass matrix $$\mathbf{M}^{-1}$$ appears everywhere in the preconditioning methods we would like to use, it would be great if this matrix was well-conditioned: as if it is, it's inverse can be very quickly and accurately approximated.
There is a condition called the inf-sup condition: if the inf-sup condition holds for the discretisation spaces used, then the mass matrix will be well-conditioned. Unfortunately, the inf-sup condition does not hold when using a combination of DP0 and P1 spaces.
All is not lost, however, as there are spaces we can use that do satisfy the inf-sup condition. We call these DUAL0 and DUAL1, and they form inf-sup stable pairs with P1 and DP0 (respectively). They are defined using the barycentric dual mesh: this mesh is defined by joining each point in a triangle with the midpoint of the opposite side, then making polygons with all the small triangles that touch a vertex in the original mesh: The mesh (left), the barycentric refinement (centre), and the dual grid (right)
Example DUAL1 and DUAL0 basis functions look like this: DUAL1 (left) and DUAL0 (right) basis functions.
For Maxwell's equations, we define BC (Buffa–Christiansen) and RBC (rotated BC) functions to make inf-sup stable spaces pairs. Example BC and RBC basis functions look like this: Example BC (left) and RBC (right) basis functions.

My thesis then gives some example Python scripts that show how these spaces can be used in Bempp to solve some example problems, concluding chapter 2 of my thesis. Why not take a break and have a slice of the following figure before reading on. An electromagnetic wave scattering off a perfectly conducting metal cake. This solution was found using a Calderón preconditioned boundary element method.

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2020-02-11
 This is the third post in a series of posts about my PhD thesis.
In the third chapter of my thesis, we look at how boundary conditions can be weakly imposed when using the boundary element method. Weak imposition of boundary condition is a fairly popular approach when using the finite element method, but our application of it to the boundary element method, and our analysis of it, is new. But before we can look at this, we must first look at what boundary conditions are and what weak imposition means.

### Boundary conditions

A boundary condition comes alongside the PDE as part of the problem we are trying to solve. As the name suggests, it is a condition on the boundary: it tells us what the solution to the problem will do at the edge of the region we are solving the problem in. For example, if we are solving a problem involving sound waves hitting an object, the boundary condition could tell us that the object reflects all the sound, or absorbs all the sound, or does something inbetween these two.
The most commonly used boundary conditions are Dirichlet conditions, Neumann conditions and Robin conditions. Dirichlet conditions say that the solution has a certain value on the boundary; Neumann conditions say that derivative of the solution has a certain value on the boundary; Robin conditions say that the solution at its derivate are in some way related to each other (eg one is two times the other).

### Imposing boundary conditions

Without boundary conditions, the PDE will have many solutions. This is analagous to the following matrix problem, or the equivalent system of simultaneous equations.
\begin{align*} \begin{bmatrix} 1&2&0\\ 3&1&1\\ 4&3&1 \end{bmatrix}\mathbf{x} &= \begin{pmatrix} 3\\4\\7 \end{pmatrix} &&\text{or}& \begin{array}{3} &1x+2y+0z=3\\ &3x+1y+1z=4\\ &4x+3y+1z=7 \end{array} \end{align*}
This system has an infinite number of solutions: for any number $$a$$, $$x=a$$, $$y=\tfrac12(3-a)$$, $$z=4-\tfrac52a-\tfrac32$$ is a solution.
A boundary condition is analagous to adding an extra condition to give this system a unique solution, for example $$x=a=1$$. The usual way of imposing a boundary condition is to substitute the condition into our equations. In this example we would get:
\begin{align*} \begin{bmatrix} 2&0\\ 1&1\\ 3&1 \end{bmatrix}\mathbf{y} &= \begin{pmatrix} 2\\1\\3 \end{pmatrix} &&\text{or}& \begin{array}{3} &2y+0z=2\\ &1y+1z=1\\ &3y+1z=3 \end{array} \end{align*}
We can then remove one of these equations to leave a square, invertible matrix. For example, dropping the middle equation gives:
\begin{align*} \begin{bmatrix} 2&0\\ 3&1 \end{bmatrix}\mathbf{y} &= \begin{pmatrix} 2\\3 \end{pmatrix} &&\text{or}& \begin{array}{3} &2y+0z=2\\ &3y+1z=3 \end{array} \end{align*}
This problem now has one unique solution ($$y=1$$, $$z=0$$).

### Weakly imposing boundary conditions

To weakly impose a boundary conditions, a different approach is taken: instead of substituting (for example) $$x=1$$ into our system, we add $$x$$ to one side of an equation and we add $$1$$ to the other side. Doing this to the first equation gives:
\begin{align*} \begin{bmatrix} 2&2&0\\ 3&1&1\\ 4&3&1 \end{bmatrix}\mathbf{x} &= \begin{pmatrix} 4\\4\\7 \end{pmatrix} &&\text{or}& \begin{array}{3} &2x+2y+0z=4\\ &3x+1y+1z=4\\ &4x+3y+1z=7 \end{array} \end{align*}
This system now has one unique solution ($$x=1$$, $$y=1$$, $$z=0$$).
In this example, weakly imposing the boundary condition seems worse than substituting, as it leads to a larger problem which will take longer to solve. This is true, but if you had a more complicated condition (eg $$\sin x=0.5$$ or $$\max(x,y)=2$$), it is not always possible to write the condition in a nice way that can be substituted in, but it is easy to weakly impose the condition (although the result will not always be a linear matrix system, but more on that in chapter 4).

### Analysis

In the third chapter of my thesis, we wrote out the derivations of formulations of the weak imposition of Dirichet, Neumann, mixed Dirichlet–Neumann, and Robin conditions on Laplace's equation: we limited our work in this chapter to Laplace's equation as it is easier to analyse than the Helmholtz equation.
Once the formulations were derived, we proved some results about them: the main result that this analysis leads to is the proof of a priori error bounds.
An a priori error bound tells you how big the difference between your approximation and the actual solution will be. These bounds are called a priori as they can be calculated before you solve the problem (as opposed to a posteriori error bounds that are calculated after solving the problem to give you an idea of how accurate you are and which parts of the solution are more or less accurate). Proving these bounds is important, as proving this kind of bound shows that your method will give a good approximation of the actual solution.
A priori error bounds look like this:
$$\left\|u-u_h\right\|\leqslant ch^{a}\left\|u\right\|$$
In this equation, $$u$$ is the solution of the actual PDE problem; $$u_h$$ is our appoximation; the $$h$$ that appears in $$u_h$$ and on the right-hand size is the length of the longest edge in the mesh we are using; and $$c$$ and $$a$$ are positive constants. The vertical lines $$\|\cdot\|$$ are a measurement of the size of a function.
Overall, the equation says that the size of the difference between the solution and our approximation (ie the error) is smaller than a constant times $$h^a$$ times the size of the solution. Because $$a$$ is positive, $$h^a$$ gets smaller as $$h$$ get smaller, and so if we make the triangles in our mesh smaller (and so have more of them), we will see our error getting smaller.
The value of the constant $$a$$ depends on the choices of discretisation spaces used: using the spaces in the previous chapter gives $$a$$ equal to $$\tfrac32$$.

### Numerical results

Using Bempp, we approximated the solution on a series of meshes with different values of $$h$$ to check that we do indeed get order $$\tfrac32$$ convergence. By plotting $$\log\left(\left\|u-u_h\right\|\right)$$ against $$\log h$$, we obtain a graph with gradient $$a$$ and can easily compare this gradient to a gradient of $$\tfrac32$$. Here are a couple of our results: The errors of a Dirichlet problem on a cube (left) and a Robin problem on a sphere (right) as $$h$$ is decreased. The dashed lines show order $$\tfrac32$$ convergence.
Note that the $$\log h$$ axis is reversed, as I find these plots easier to interpret this way found. Pleasingly, in these numerical experiments, the order of convergence agrees with the a priori bounds that we proved in this chapter.

These results conclude chapter 3 of my thesis. Why not take a break and refill the following figure with hot liquid before reading on. The solution of a mixed Dirichlet–Neumann problem on the interior of a mug, solved using weakly imposed boundary conditions.

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2020-02-13
 This is the fourth post in a series of posts about my PhD thesis.
The fourth chapter of my thesis looks at weakly imposing Signorini boundary conditions on the boundary element method.

### Signorini conditions

Signorini boundary conditions are composed of the following three conditions on the boundary:
\begin{align*} u &\leqslant g\\ \frac{\partial u}{\partial n} &\leqslant \psi\\ \left(\frac{\partial u}{\partial n} -\psi\right)(u-g) &=0 \end{align*}
In these equations, $$u$$ is the solution, $$\frac{\partial u}{\partial n}$$ is the derivative of the solution in the direction normal to the boundary, and $$g$$ and $$\psi$$ are (known) functions.
These conditions model an object that is coming into contact with a surface: imagine an object that is being pushed upwards towards a surface. $$u$$ is the height of the object at each point; $$\frac{\partial u}{\partial n}$$ is the speed the object is moving upwards at each point; $$g$$ is the height of the surface; and $$\psi$$ is the speed at which the upwards force will cause the object to move. We now consider the meaning of each of the three conditions.
The first condition ($$u \leqslant g$$ says the the height of the object is less than or equal to the height of the surface. Or in other words, no part of the object has been pushed through the surface. If you've ever picked up a real object, you will see that this is sensible.
The second condition ($$\frac{\partial u}{\partial n} \leqslant \psi$$) says that the speed of the object is less than or equal to the speed that the upwards force will cause. Or in other words, the is no extra hidden force that could cause the object to move faster.
The final condition ($$\left(\frac{\partial u}{\partial n} -\psi\right)(u-g)=0$$) says that either $$u=g$$ or $$\frac{\partial u}{\partial n}=\psi$$. Or in other words, either the object is touching the surface, or it is not and so it is travelling at the speed caused by the force.

### Non-linear problems

It is possible to rewrite these conditions as the following, where $$c$$ is any positive constant:
$$u-g=\left[u-g + c\left(\frac{\partial u}{\partial n}-\psi\right)\right]_-$$
The term $$\left[a\right]_-$$ is equal to $$a$$ if $$a$$ is negative or 0 if $$a$$ is positive (ie $$\left[a\right]_-=\min(0,a)$$). If you think about what this will be equal to if $$u=g$$, $$u\lt g$$, $$\frac{\partial u}{\partial n}=\psi$$, and $$\frac{\partial u}{\partial n}\lt\psi$$, you can convince yourself that it is equivalent to the three Signorini conditions.
Writing the condition like this is helpful, as this can easily be added into the matrix system due to the boundary element method to weakly impose it. There is, however, a complication: due to the $$[\cdot]_-$$ operator, the term we add on is non-linear and cannot be represented as a matrix. We therefore need to do a little more than simply use our favourite matrix solver to solve this problem.
To solve this non-linear system, we use an iterative approach: first make a guess at what the solution might be (eg guess that $$u=0$$). We then use this guess to calculate the value of the non-linear term, then solve the linear system obtained when the value is substituted in. This gives us a second guess of the solution: we can do follow the same approach to obtain a third guess. And a fourth. And a fifth. We continue until one of our guesses is very close to the following guess, and we have an approximation of the solution.

### Analysis

After deriving formulations for weakly imposed Signorini conditions on the boundary element method, we proceed as we did in chapter 3 and analyse the method.
The analysis in chapter 4 differs from that in chapter 3 as the system in this chapter is non-linear. The final result, however, closely resembles the results in chapter 3: we obtain and a priori error bounds:
$$\left\|u-u_h\right\|\leqslant ch^{a}\left\|u\right\|$$
As in chapter 3, The value of the constant $$a$$ for the spaces we use is $$\tfrac32$$.

### Numerical results

As in chapter 3, we used Bempp to run some numerical experiments to demonstrate the performance of this method. The error of our approximate solutions of a Signorini problem on the interior of a sphere with meshes with different values of $$h$$ for two choices of combinations of discrete spaces. The dashed lines show order $$\tfrac32$$ convergence.
These results are for two different combinations of the discrete spaces we looked at in chapter 2. The plot on the left shows the expected order $$\tfrac32$$. The plot on the right, however, shows a lower convergence than our a priori error bound predicted. This is due to the matrices obtained when using this combination of spaces being ill-conditioned, and so our solver struggles to find a good solution. This ill-conditioning is worse for smaller values of $$h$$, which is why the plot starts at around the expected order of convergence, but then the convergence tails off.

These results conclude chapter 4 of my thesis. Why not take a break and snack on the following figure before reading on. The solution of a mixed Dirichlet–Signorini problem on the interior of an apple, solved using weakly imposed boundary conditions.

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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.

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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.

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.

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