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In this work we consider the long-time behaviour of dissipative evolution equations; in particular, the construction of Approximate Inertial Manifolds (AIMs) for the Ginzburg-Landau equation (GLE) and their subsequent application to the creation of more efficient and more accurate numerical schemes. In the first part, the time analyticity for the solutions of a class of dissipative evolution equations is shown. This class includes Reaction-Diffusion, GLE, Navier-Stokes, and Cahn-Hilliard equations. Existing methods are generalized and extended, and it is shown that the solutions of these equations have a unique analytic extension to an infinite pencil-shaped domain about the positive real axis in the complex plane. Moreover, in the space-periodic case, the solutions are shown to be in a Gevrey class, to be${\\cal C}\\sp\\infty$in the space variable, and to have exponential decay of the Fourier coefficients. In the second part, the preceding result is applied to develop a new method of construction of AIMs which produces an infinite series of increasingly higher order AIMs for the GLE and associates with each a thin neighborhood into which the orbits enter in finite time and with exponential speed. These manifolds are a substitute for Inertial Manifolds when the existence of the Inertial Manifold is not known and are shown to localize the universal attractor in the phase space. The method of construction is general and can be applied to other equations. Finally, in the third part, using the explicit non-linear equations of the first two nontrivial AIMs, two new numerical schemes are implemented for the GLE, as well as a traditional, linear Galerkin scheme. Comparisons of the accuracy of these three schemes are made, showing gains in stability and accuracy.

Reductions of the self-consistent mean field theory model of amphiphilic molecules in solvent can lead to a singular family of functionalized Cahn-Hilliard energies. We modify these energies, mollifying the singularities to stabilize the computation of the gradient flows and develop a series of benchmark problems that emulate the \"morphological complexity\" observed in experiments. These benchmarks investigate the delicate balance between the rate of absorption of amphiphilic material onto an interface and a least energy mechanism to disperse the arriving mass. The result is a trichotomy of responses in which two-dimensional interfaces either lengthen by a regularized motion against curvature, undergo pearling bifurcations, or split directly into networks of interfaces. We evaluate a number of schemes that use second order BDF2-type time stepping coupled with Fourier pseudo-spectral spatial discretization. The BDF2-type schemes are either based on a fully implicit time discretization with a PSD nonlinear solver, or upon IMEX, SAV, ETD approaches. All schemes use a fixed local truncation error target with adaptive time-stepping to achieve the error target. Each scheme requires proper \"preconditioning\" to achieve robust performance that can enhance efficiency by several orders of magnitude.