In this thesis we revisit theoretical background for space­time boundary element methods for the heat equation and its implementation. We restrict ourselves to solving the one­, and two dimensional Dirichlet heat equation. A new approach is proposed to approximate the Galerkin matrix entries in a semianalytical fashion, requiring a reduced order of quadrature. This method can be applied to non­uniform meshes, but is restricted to right­triangular meshes. Using this approach, a system of Galerkin equations is created and solved iteratively with the use of the Generalised Residual Method (GMRES). Operator preconditioners and preconditioners originating from domain decomposition methods are summarised and implemented for a two dimensional Dirichlet problem. In the case of operator preconditioning, a diagonal duality pairing proposed by Stevenson and van Venetië [23] is used in the implementations. The Restricted Additive Schwarz method is considered as both a preconditioner and a basic iterative method. The Calderón preconditioner, an operator preconditioner, is used to test the efficiency of parallel preconditioned GMRES implementations, as this preconditioner provides a dense matrix. For different amount of processes, the parallel GMRES implementations are investigated. Using row­wise decomposition, parallel GMRES becomes increasingly time­efficient, as the level of refinement increases. However, the Induced Dimension Reduction method, a different non­symmetric solver, currently outperforms the parallel GMRES implementation.

Janus particles are colloidal particles for which one half of the surface has different attributes than the other half. One property of a spherical dielectric particle with half of its surface covered by a layer of another dielectric or metal is that it has a non-uniform scattering pattern when exposed to light. However, the angle with which the light is shone on the particle has a large effect on the scattering pattern produced. Thus it is important that we are able to orient these Janus particles. The orientation can be controlled if we apply an electric field to the particle for example. The movement of colloidal particles with an electric field is widely studied and this field is called dielectrophoresis. For a Janus particle, the calculations for the force and torque become complicated. The movement and rotation of these particles have been studied, however, no analytic solution has been found. In this report, we derive a semi-analytic description of the force and torque due to an external electric field on a spherical Janus particle. For this, first the potential due to an external electric field is determined and then the force and torque are calculated with two methods: the dipole approximation and the Maxwell Stress Tensor method. In the dipole approximation, there is no force on the Janus particle. But, there is a torque on the particle in the dipole approximation. Due to this torque, the Janus particle will orient itself such that its cap points in the direction perpendicular to the applied field. For the torque calculated with the Maxwell Stress Tensor, we get a similar result as in the dipole approximation. On the other hand, according to the calculations with the stress tensor, there is a relatively small force on the particle.

This thesis is about the wave equation. The wave equation describes waves that propagate through a certain medium. The solution of the wave equation is a mathematical description of what that wave looks like. There are many fields of study where the wave equation is used to model processes that behave like travelling waves. For instance, a vibrating string or membrane can be modelled by the wave equation very well. Furthermore, the wave equation can be used to model light or sound waves and how they reflect and refract when travelling through different materials. Because the wave equation is such an important tool to model these phenomena, there are many people working on solving the wave equation in different contexts, i.e. finding a solution. However, it is difficult (often impossible) to find an exact solution. Therefore, people usually calculate a ’solution’ that is approximately correct. An important question to ask is: ’Does the wave equation always have a solution?And is there only one solution?’. Physically, the answer is clear. If a string is vibrating, it clearly cannot vibrate in two ways at the same time and it will always vibrate in some particular way. Even though the answer is clear physically, answering this question mathematically is more difficult. It is good to remember that the wave equation is just a mathematical model of propagating waves, and it could very well be that the wave equation has more than one solution or no solution at all in some particular context. The answer of this question is important for the people calculating approximate solutions to the wave equation. If there does not exist a unique solution, the process of calculating the approximate solution may break down. Answering that question is the subject of this thesis. We will first introduce the necessary mathematical tools, after which we will use existing research to find out under which conditions the wave equation has one, and only one solution. Finally, we will extend our results to more general equations than just the wave equation.

We consider the game cops and robbers, which is a pursuit-evasion game played on a graph G. The cops and the robber take turns moving across the vertices of G, where the goal for the cops is to eventually catch the robber. Specifically, we study the cop number of G, i.e. the minimum number of cops that is needed to catch the robber on G. We investigate the relation between the cop number of the Erdős-Rényi (ER) random graph G(n, p_n) and compare it to another graph parameter, the domination number. Our goal is to collect results from previous research and create an overview. Throughout this thesis, we provide some examples of graphs for which the cop number is equal to the domination number. Furthermore, we provide proofs for some results on the domination number of the ER random graph. The main takeaway is that the cop number is asymptotic to the domination number for particular values of p_n.

Differential equations are commonly solved by, first, discretising the domain and then using an iterative method on the resulting system of equations. Refining the mesh gives a more accurate solution. However, the iterative method does not necessarily converge quickly to a good approximate solution anymore. To deal with this issue, we can add a preconditioner to the system. A good preconditioner enhances the speed of convergence. Two ways of finding a good preconditioner are known. The first uses the properties of the matrix of the system, that is an algebraic preconditioner. The second uses the properties of the differential equation that give rise to the system, which is an operator preconditioner. This thesis makes a connection between both types of preconditioning. First, we discretise a differential equation and perform numerical test on the system to see if the algebraic preconditioning works. Then, the domains on which the matrix and preconditioner act are defined in terms of the differential equations. We conclude that the matrix containing the differential operators acts on H(div, Ω) × H1(Ω) and the preconditioner acts on [L2(Ω)]^2× H2(Ω). So we need to constrain the domains in such a way that the operators both act on the same domain: that is, H(div, Ω) × H2(Ω). If the function is in this domain,then we know that the algebraic preconditioner is an effective preconditioner.

Differential equations are commonly solved by, first, discretising the domain and then using an iterative method on the resulting system of equations. Refining the mesh gives a more accurate solution. However, the iterative method does not necessarily converge quickly to a good approximate solution anymore. To deal with this issue, we can add a preconditioner to the system. A good preconditioner enhances the speed of convergence. Two ways of finding a good preconditioner are known. The first uses the properties of the matrix of the system, that is an algebraic preconditioner. The second uses the properties of the differential equation that give rise to the system, which is an operator preconditioner. This thesis makes a connection between both types of preconditioning. First, we discretise a differential equation and perform numerical test on the system to see if the algebraic preconditioning works. Then, the domains on which the matrix and preconditioner act are defined in terms of the differential equations. We conclude that the matrix containing the differential operators acts on H(div, Ω) × H1(Ω) and the preconditioner acts on [L2(Ω)]^2× H2(Ω). So we need to constrain the domains in such a way that the operators both act on the same domain: that is, H(div, Ω) × H2(Ω). If the function is in this domain,then we know that the algebraic preconditioner is an effective preconditioner.