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In this paper we consider the numerical approximation of elliptic problems by spectral methods in domains subdivided into substructures. We review an iterative procedure with interface relaxation, reducing the given differential problem to a sequence of Dirichlet and mixed Neumann-Dirichlet problems on each subdomain. The iterative procedure is applied to both tau and collocation spectral approximations. Two optimal strategies for the automatic selection of the relaxation parameter are indicated. We present several numerical experiments showing the convergence properties of the iterative scheme with respect to the decomposition. A multilevel technique for domain decomposition methods is proposed. [ABSTRACT FROM AUTHOR]

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