Georgia Institute of Technology College of Sciences

The computation of invariant manifolds for parabolic PDE is an important problem due to its many applications. One of the main difficulties is dealing with irregular high dimensional domains when the classical Fourier methods are not applicable, and it is necessary to employ more sophisticated numerical methods. This work combines the parameterization method based on an invariance equation for the invariant manifold, with the finite element method. We implement the method for a variety of example problems having both polynomial and non-polynomial nonlinearities, on non-convex two-dimensional polygonal domains (not necessary simply connected), for equilibrium solutions. We implement a-posteriori error indicators which provide numerical evidence of the accuracy of the computations. This is a joint work with J.D Mireles-James, and Necibe Tuncer.