Efficient Spectral-Galerkin Algorithms for Direct Solution of Second-Order Equations Using Ultraspherical 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 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 N d+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. AMS subject classifications. 65N35, 65N22, 65F05, 35J05PII. S1064827500378933 1. Introduction. This paper aims to develop some efficient spectral algorithms based on the ultraspherical-Galerkin methods (UGM) for elliptic second-order differential equations in one and two space variables.Spectral methods (see, for instance, Canuto et al.[2], Doha [3,4,5], and Gottlieb and Orszag [8]) involve representing the solution to a problem in terms of a truncated series of smooth global functions. They give very accurate approximations for a smooth solution with relatively few degrees of freedom. For Dirichlet problems, Heinrichs [10] uses the Chebyshev polynomials as basis functions. It turns out that for the well-known standard spectral methods (tau, Galerkin, collocation) the condition number is very large and grows as O(N 4 ) (N is the number of retained modes of the approximation; see Orszag [14]). Heinrichs [11] proposes a spectral method based on a subclass of orthogonal ultraspherical polynomials (λ = 3 2 ) for solving the Helmholtz equation in two dimensions subject to Dirichlet homogeneous boundary conditions with a symmetric and sparse matrix, whose condition number grows only as O(N 2 ); and he shows in [12] that certain algebraic spectral multigrid methods can be efficiently used for solving the resulting system.In this paper we are concerned with the direct solution techniques for the secondorder elliptic equations, using the ultraspherical-Galerkin approximations. We present appropriate bases for the UGM applied to Helmholtz elliptic equations with various boundary conditions. This leads to discrete systems with specially structured matrices that can be efficiently inverted. We note that the improved technique of Heinrichs [10] and the two efficient Legendre-Galerkin and Chebyshev-Galerkin approximations developed by Shen [15,16], respectively, and some other very interesting cases, can be obtained directly as special cases from our proposed ultraspherical-Galerkin approximations. This motivates our interest for making such a generalization.

A spectral method for elliptic equations: the Neumann problem

“…with 1 and 2 defined as in (7)(8). As before, the Lax-Milgram Theorem implies the existence of a unique solution u to (30) with…”

A spectral method for elliptic equations: the Dirichlet problem

“…From the more recent literature, we cite [5,[7][8][9] and [19]. Their bibliographies contain references to earlier papers on spectral methods.…”

Jacobi spectral Galerkin method for elliptic Neumann problems

“…This equation can be solved in particular by a double diagonalization method described in [2], which is very similar to that of the Dirichlet spectral solver [1]. Also, it can be solved by the matrix decomposition method described in [5], which is better known in the field of spectral methods as the matrix diagonalization method [10,12]. Throughout this paper we use the symbol ⊗ to denote both the tensor product of matrices and the tensor product of function spaces.…”