document: Single and multi-interval Legendre tau - methods in time for parabolic equations In this paper , we take the parabolic equation with periodic boundary conditions as a model to present a spectral method with the Fourier approximation in spatial and single/multi-interval Legendre Petrov-Galerkin methods in time . For the single interval spectral method in time , we obtain the optimal error estimate in L/sup 2 / - norm . For the multi-interval spectral method in time , the L/sup 2 / - optimal error estimate is valid in spatial . Numerical results show the efficiency of the methods
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