Coupled Harmonic Equations , SOR , and Chebyshev Acceleration

A coupled pair of harmonic equations is solved by the application of Chebyshev acceleration to the Jacobi, Gauss-Seidel, and related iterative methods, where the Jacobi iteration matrix has purely imaginary (or zero) eigenvalues. Comparison is made with a block SOR method used to solve the same problem. Introduction. In [4], we proposed a general block SOR method for solving the biharmonic equation as a coupled set of finite-difference equations. Here, we consider related methods and compare them to the SOR method. The methods considered here are Chebyshev accelerated Jacobi and Gauss-Seidel as well as others, the cyclic Chebyshev semi-iteration method [5], and the unsymmetric modified SOR method [20], [21]. It is shown, by comparing spectral radii, that the SOR method of [4] is at least as fast as any of the above methods. The analysis applies to a block cyclic matrix of index 2, or one that can be written in the form (2.1). The interesting feature of the analysis is that the Jacobi matrix has purely imaginary (or zero) eigenvalues, whereas most previous work assumes real eigenvalues, or some complex. To fix notation, consider the iterative process (0.1) = tVn> + d, where there exists a vector u such that (0.2) u = Gu + d. Let X,(G) be an eigenvalue of G and let S(G) = X = max; |X,(G)| be the spectral radius of G. The rate of convergence of the iterative process (0.1) is defined by (0.3) R(G) = -In S(G) = -In X, when X < 1 [18]. In the event that G is a function of n, i.e., (0.4) = Gn«(n) + d, we define the average rate of convergence as [16] (0.5) R(Gn) = -(In S(G„))/n Received April 27, 1971, revised October 14, 1971. A MS 1970 subject classifications. Primary 65F10, 65N20, 15A06.