This chapter focuses on the project of finding the potential for a given distribution of charges in a two-dimensional system, which does not possess any symmetrical properties, an extension of the cylindrical potential problem discussed in the previous chapter. Using a method of minimising a functional, specifically the Gauss–Seidel method of iterative minimisation, the Poisson’s equation is adjusted to a 2D case, neglecting one partial derivative in Cartesian coordinates. We subsequently derive a discretised form of the functional, leading to a multi-variable function, following which the problem can be solved using the Gauss–Seidel iterative method. The numerical method discussed here is the finite elements method (FEM), with an emphasis on the need for a specific sequence for updating values to optimise computation efficiency. The discussion sheds light on the importance of the uniqueness of solutions in electrostatic systems, thereby exploring a fundamental question in electrostatics. The concluding part of the chapter provides an outline of a numerical algorithm for the problem, suggesting potential modifications and points for further exploration.

The application of finite-difference methods to boundary-value problems is considered using the Poisson equation as a model problem.Direct and iterative methods are given that are effective for solving elliptic partial differential equations in multidimensions having various types of boundary conditions.Multigrid methods are given particular attention given their generality and efficiency.Treatment of nonlinear terms are illustrated using Picard and Newton linearization.

Computational linear algebra builds on the methods in Part I for solving systems of linear algebraic equations and the algebraic eigenproblem appropriate for small systems to methods amenable to approximate computer solutions for large systems.These include direct and iterative methods for solving systems of equations, such as LU decomposition and Gauss-Seidel iteration.A popular algorithm based on QR decomposition is described for solving large algebraic eigenproblems for the full spectrum of eigenpairs, and the Arnoldi method for a subset of eigenpairs of sparse matrices.