Nyström Method

In this project, we explore how one can solve partial differential equations (PDEs) using boundary integral methods. These methods are particularly effective for solving boundary value problems for elliptic PDEs in domains with parametrisable boundaries. The essence of the methods is to use the fundamental solutions of the governing equations (i.e. the domain independent Green’s functions) to formulate the solution as a layer potential integral over the domain boundary.

The key advantage of boundary integral methods is that they reduce the dimension of the problem by one. This is because we are transforming the problem from something involving the entire domain to something involving just the boundary. This is hugely benficial in simplifying program implementations and in reducing computation costs.

This notebook will serve as an introduction to solving the Laplace equation using the double layer potential formulation. We shall go about this using a boundary integral method known as the Nyström Method. We will discuss the theoretical foundation, the implementation details, and how to solve the interior Dirichlet problem for the Laplace equation in smooth domains as a concrete example. Although this is a fairly simple example the work demonstrated can be extended to other elliptic equations, such as the Helmholtz equation or biharmonic equations, or more general PDE solvers.