The partial differential equations (PDEs) that arise in applications can only rarely be solved in closed form. Even when they can be, the solutions are often impractical to work with and to visualize. Numerical techniques, on the other hand, can be applied successfully to virtually all well-posed PDEs. Broadly applicable techniques include finite element (FE), finite volume (FV), finite difference (FD), and, more recently, spectral methods. The complexity of the domain and the required levels of accuracy are often the key factors in selecting among these approaches.

Finite-element methods are particularly well suited to problems in very complex geometries (e.g. 3-D engineering structures), whereas spectral methods can offer superior accuracies (and cost efficiencies) mainly in simple geometries such as boxes and spheres (which can, however, be combined into more complex shapes). FD methods perform well over a broad range of accuracy requirements and (moderately complex) domains.

Both FE and FV methods are closely related to FD methods. FE methods can frequently be seen as a very convenient way to generate and administer complex FD schemes and to obtain results with relatively sharp error estimates. The connection between spectral methods – in particular the so-called pseudospectral (PS) methods, the topic of this book – and FD methods is closer still. A key theme in this book is to exploit this connection, both to make PS methods more intuitively understandable and to obtain particularly powerful and flexible PS variations.