Discovering spectral methods

Spectral methods, as a technique for discretizing partial differential equations, were introduced almost half a century ago in a series of papers by Steve Orszag (see for example ) and in . They were given a major boost by David Gottlieb and Steve Orszag's book where the foundations were laid for their generalization to approximations other than Fourier series periodic functions. They are characterized by the following two points:

• approximation by high-degree polynomials ;

• the use of tensorized polynomial bases.

Another aspect of this evolution is that the use of the fast Fourier transform algorithm, which was important just 15 years ago, has since been relegated to the background by increased computing power and improved matrix multiplication algorithms.

Like many other discretizations today, spectral methods make use of the variational formulation of the initial problem, and the discrete problem is most often constructed by Galerkin's method, so that the error between the exact solution and the approximated solution is of the same order as the best approximation error in discrete space. Approximation by high-degree polynomials leads to an infinite-order discretization, in the following sense: if N denotes the degree of the polynomials used in the discretization, the a priori error behaves as N ^-σ , where σ can be as large as the regularity of the solution allows. Using tensorized bases leads to mathematical and, above all, numerical simplification: indeed, the cost of the matrix-vector product in the implementation of these methods is greatly reduced, which lowers the computational cost. The corresponding disadvantage is that the basic domains for spectral methods are tensorized (a rectangle...