  This paper develops validated computational methods for studying infinite
dimensional stable manifolds at equilibrium solutions of parabolic PDEs,
synthesizing disparate errors resulting from numerical approximation. To
construct our approximation, we decompose the stable manifold into three
components: a finite dimensional slow component, a fast-but-finite dimensional
component, and a strongly contracting infinite dimensional "tail". We employ
the parameterization method in a finite dimensional projection to approximate
the slow-stable manifold, as well as the attached finite dimensional invariant
vector bundles. This approximation provides a change of coordinates which
largely removes the nonlinear terms in the slow stable directions. In this
adapted coordinate system we apply the Lyapunov-Perron method, resulting in
mathematically rigorous bounds on the approximation errors. As a result, we
obtain significantly sharper bounds than would be obtained using only the
linear approximation given by the eigendirections. As a concrete example we
illustrate the technique for a 1D Swift-Hohenberg equation.
