Abstract: Traditionally, stationary solutions of PDEs have been studied numerically via Galerkin projections and continuation algorithms. Many arguments may be made as to why this approach is expected to yield useful results under certain conditions. However, the traditional approach does not include verifying that the required conditions are met and may lead to misleading results. In this talk, we present a rigorous numerical method which combines topological tools with continuation algorithms to obtain proofs about the structure of the set of stationary solutions at roughly the same cost as the traditional continuation approach. Furthermore, the constructive nature of these techniques may be used to detect when and where further computations are required to uncover the desired structure. We will study these techniques applied to the Swift-Hohenberg equation and, if time permits, discuss how these and similar computations can be used to construct a model of the global attractor for the system.

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