### Fast tensor-product solvers: Part II: Spectral discretization in space and time

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**Abstract:** We consider the numerical solution of an unsteady convection-diffusion equation using high order polynomial approximations both in space and time. General boundary conditions and initial conditions can be imposed. The method is fully implicit and enjoys exponential convergence in time and space for analytic solutions. This is confirmed by numerical experiments (in one space dimension) using a spectral element approach in time and a pure spectral method in space. A fast tensor-product solver has been developed to solve the coupled system of algebraic equations for the *O(N*^{d}) unknown nodal values within a single space-time spectral element. This solver has a fixed complexity of *O(N*^{d+1}) floating point operations and a memory requirement of *O(N*^{d}) floating point numbers. An alternative solution method, allowing for a parallel implementation and more easily extendable to more complex problems is sketched out at the end of the paper.

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**Date:** 2007-09-24