ABSTRACT
Eu = 0 on a domain S, (11.1a)
subject to boundary conditions
Bu = 0 on ∂S, (11.1b)
where ∂S is the boundary of the domain S, are a natural progression of the work of Lanczos (1938) and Clenshaw (1957) on ordinary differential equations. However, the first formal publications in the topic of PDEs appear to be those of Elliott (1961), Mason (1965, 1967) and Fox & Parker (1968) in the 1960s, where some of the fundamental ideas for extending onedimensional techniques to multi-dimensional forms and domains were first developed. Then in the 1970s, Kreiss & Oliger (1972) and Gottlieb & Orszag (1977) led the way to the strong development of so-called pseudo-spectral methods, which exploit the fast Fourier transform of Cooley & Tukey (1965), the intrinsic rapid convergence of Chebyshev methods, and the simplicity of differentiation matrices with nodal bases.
