An efficient estimate based on FFT in topological verification method

In this paper, localized patterns of the quintic Swift–Hohenberg equation are studied. A numerical verification method with the Conley index theory developed in Zgliczyński and Mischaikow [Rigorous numerics for partial differential equations: the Kuramoto–Sivashinsky equation, Found. Comput. Math. 1 (2001) 255–288] is used in order to prove these patterns. A new technique to efficiently obtain estimates for nonlinear terms is presented. The key idea is based on the pseudo-spectral method. It is shown that this technique is inevitable for the verification of the localized patterns.