This chapter discusses the application of the variational principle and finite element (FE) methods to electrostatic systems, using a cylindrical capacitor as a representative example. The variational principle, which identifies the solution that minimises a system’s total energy, provides a foundation for numerical techniques such as the FE methods. These methods offer flexibility in the selection of points in the independent variable space, which can be adapted according to the expected function behaviour. The system of focus is a capacitor comprising two coaxial metallic cylinders, chosen for its high symmetry and real-world relevance in areas like telecommunication cables. The chapter presents the Poisson’s equation, demonstrates the variational principle, and introduces the FE method. A numerical example of a cylindrical capacitor is also provided, illustrating how to convert the problem into one of functional minimisation using Poisson’s equation and then solve it using the FE method. The Gauss–Seidel iterative minimisation procedure is used in the solution process. The discussion provides a foundation for extending these methods to 2D and 3D systems in subsequent chapters.

This chapter covers the problem of a single quantum well filled with electrons, specifically, the changes that occur when electrons are introduced into an empty quantum well. Utilising the ’jellium’ model, the chapter commences by identifying the energy levels of an empty quantum well composed of infinite dipole planes filled with positively charged jellium. The subsequent introduction of electrons leads to a significant change in the well’s shape due to interactions within a non-uniform electron gas. However, the complexity of this problem is simplified by considering only Hartree interactions, allowing for self-consistent calculations. The density of the negative charge introduced is determined, then added to the uniform density of the jellium, resulting in a new potential energy shape of the well. The iterative process of determining new energy levels, populating them with electrons, and re-evaluating continues until a chosen property converges. The chapter concludes by demonstrating how to model the potential of an empty well and how this potential changes in a self-consistent procedure when the well is filled with electrons.

This project explores the boundary value problem (BVP) for ordinary differential equations concerning the gravitational field inside a star. The study equates the problem to the electric field inside an atom and reduces the partial differential equation of Poisson’s type to a second order ordinary differential equation, utilising high symmetry. The uniqueness of the solution is ensured by applying two conditions at two ends of the independent variable range. The Numerov’s (Cowell’s) algorithm is employed to solve the equation accurately. However, it is identified that numerical solutions can be very sensitive to the value chosen for the second point, necessitating a recursive scheme. The project also introduces the application of Gauss law, Poisson’s equation, and the Numerov–Cowells algorithm in determining the gravitational potential inside a star given a model radial mass density distribution. The study concludes by discussing the possibility of treating the recursive formula as a tridiagonal system of linear equations and solving it with Gaussian elimination with backward substitution algorithm.

Unlike in Chapter 5, this project aims at finding a real mass density distribution of a hydrogen star of given mass. For that purpose an equilibrium condition for the gravitational and pressure-induced forces acting on a mass element is utilised. Using the integral form of Gauss’s law and the equation of state, we establish an integro-differential equation describing the mass density distribution. To numerically solve the integro-differential equation, we adapt the Adams–Bashforth method and implement a linear extrapolation based on known data points. This approach involves modelling the star as a gas under pressure using an exponential form for the equation of state, which helps in avoiding gravitational collapse. The equation of state is derived based on density functional theory data. We also discuss the constraints of this model and the significance of the parameters within it. The chapter concludes by suggesting potential numerical experiments to examine the influence of these parameters and their physical interpretation. This analysis aims to provide a more comprehensive understanding of stellar structure and the behaviour of mass density distribution within stars.

Chapter 8 starts out with a physics motivation, as well as a mathematical statement of the problems that will be tackled in later sections. Starting from differential-equation initial-value problems, the text introduces both explicit and implicit methods, like backward Euler and the fourth-order Runge-Kutta method. Emphasis is placed on the interplay between method stability and problem conditioning (stiffness). The chapter then discusses boundary-value problems, first, via a combination of the earlier machinery on initial-value problems along with root-finding techniques and, second, via a finite-difference/matrix approach, which converts the problem to a linear system of equations. Next, the chapter tackles eigenvalue problems, again, via either rootfinding plus earlier tools, or a finite-difference approach; this time, the latter turns into a matrix eigenvalue problem. The second edition discussesfinite-difference approaches to solving the diffusion equation. The chapter is rounded out by a physics project, on Poisson’s equation in two dimensions, and a problem set. The physics project introduces and uses the two-dimensional fast Fourier transform, as part of a spectral method applied to the solution of a partial differential equation.