Efficient Spectral-Galerkin Algorithms for Direct Solution of Second-Order Equations Using Ultraspherical Polynomials

It is well known that spectral methods (tau, Galerkin, collocation) have a condition number of O(N4) (N is the number of retained modes of polynomial approximations). This paper presents some efficient spectral algorithms, which have a condition number of O(N2), based on the ultraspherical-Galerkin methods for second-order elliptic equations in one and two space variables. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to systems with specially structured matrices that can be efficiently inverted. The complexities of the algorithms are a small multiple of Nd+1 operations for a d-dimensional domain with (N-1)d unknowns, while the convergence rates of the algorithms are exponential for problems with smooth solutions.

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