Spectral methods solve elliptic PDEs numerically with errors bounded by an exponentially decaying function of the number of modes when the solution is analytic. For time dependent PDEs, almost all focus has been on low-order finite difference schemes for the time derivative and spectral schemes for spatial derivatives. This is not ideal since the time discretization error

destroys the spectral convergence in space. Space-time spectral methods are new methods which apply spectral discretization in both space and time. This is a series talk and consists of two parts.

Part-I: in the first talk, we report on exponential convergence of two such methods for the heat equation. We show that the condition number of the methods can be shown to be O(n^4), where n is the number of spectral modes in each direction.