This thesis investigated the influence of matrix structure, arising from various boundary conditions, on the performance of both direct and iterative solvers for linear systems resulting from discretized partial differential equations (PDEs). Using the negative one-dimensional Poisson equation as a model problem, system matrices were constructed for Dirichlet, Neumann, and mixed boundary conditions, including symmetrized variants.

The study classified the resulting matrices in terms of singularity and positive definiteness. Direct solvers such as LU decomposition and Cholesky factorization were tested for accuracy and applicability. Iterative solvers (Jacobi, Gauss-Seidel, and Successive Over-Relaxation or SOR) were analysed using the spectral radius of their iteration matrices to predict convergence.

Key findings include:
- Matrices from Dirichlet and mixed boundary conditions were non-singular; Neumann matrices were singular but still allowed convergence in iterative solvers under specific conditions.
- Cholesky factorization only applied to positive definite systems, while LU decomposition was more generally applicable to non-singular systems.
- Gauss-Seidel converged faster than Jacobi, with twice the convergence speed in tridiagonal cases.
- SOR achieved the fastest convergence when using the optimal relaxation parameter, though this optimal ω is generally not known a priori.
- Even when the spectral radius ρ(B) approached 1, as in Neumann systems, convergence was observed due to the structure of the eigenvalue spectrum.
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The thesis concludes that the structural properties of the matrix, such as singularity and diagonal dominance, are critical in determining the suitability and efficiency of numerical solvers. It also highlights the fragility of these methods when applied to nearly singular systems, motivating future research into more robust solution techniques and convergence criteria.

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.

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.

In this report, a model is presented to alleviate some of the computational work that goes into the effort of finding the magnetic properties of magnetocaloric materials. The model utilizes an interior point optimization routine to solve for the minimal exchange energy configuration of a system, given the exchange interactions of the material. The model is tested against four materials (Ni, MnO, Fe\textsubscript{2}P and Mn\textsubscript{2}Sb). For Ni and MnO, the exchange interactions are also computed. Three iterations of the model are compared. The base model, which only considers exchange interactions inside a chosen supercell, the base model with the inclusion of boundary conditions, and the base model with boundary conditions and the addition of an algorithm to find optimal solutions. \\ The algorithm analyzes the found results by the optimization routine, and if the result is considered not properly symmetric, runs the optimization routine another time, from a symmetrical starting point obtained from the outcome of the previous run. \\ In all versions of the model, effectiveness (percent of runs that resulted in the optimal configuration) and average run times were recorded. Three initialization methods for the model were used, and also tested for their effectiveness. For the algorithm, a parameter $\gamma$ is introduced that changes the size of some of the moments for the new starting points. Six different values for $\gamma$ were tested for their effectiveness against a test set of suboptimal solutions.
The model with the addition of boundary conditions and the algorithm performed the best out of the three iterations of the model, with an effectiveness of 99.895\%, and an average run time ranging from 0.62 s for 2$\times$2$\times$2 Ni, to 94.64 s for 3$\times$3$\times$3 Fe\textsubscript{2}P, in the case of $\gamma = 0.3$. To conclude, the model with the inclusion of the boundary conditions and the algorithm proves to be a robust method to evaluate the magnetic configuration of a material, especially for smaller systems.

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.