A Nonstandard Cyclic Reduction Method, Its Variants and Stability

Abstract

A nonstandard cyclic reduction method is introduced for solving the Poisson equation in rectangular domains. Different ways of solving the arising reduced systems are considered. The partial solution approach leads to the so-called partial solution variant of the cyclic reduction (PSCR) method, while the other variants are obtained by using the matrix rational polynomial factorization technique, including the partial fraction expansions, the fast Fourier transform (FFT) approach, and the combination of Fourier analysis and cyclic reduction (FACR) techniques. Such techniques have originally been considered in the standard cyclic reduction framework. The equivalence of the partial solution and the partial fraction techniques is shown. The computational cost of the considered variants is ${\mathcal O} (N\log N)$ operations, except for the FACR techniques for which it is ${\mathcal O} (N\log\log N)$. The stability estimate for the considered method is constructed, and the stability is demonstrated by numerical experiments.