Efficient spectral-Galerkin algorithms for direct solution for second-order differential equations using Jacobi polynomials

Abstract

It is well known that spectral methods (tau, Galerkin, collocation) have a condition number of $O(N^{4})$ ( $N$ is the number of retained modes of polynomial approximations). This paper presents some efficient spectral algorithms, which have a condition number of $O(N^{2})$ , based on the Jacobi–Galerkin methods of 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 $N^{d+1}$ operations for a $d$ -dimensional domain with $(N-1)^d$ unknowns, while the convergence rates of the algorithms are exponentials with smooth solutions.