Chebyshev type spectral methods are widely used in numerical simulations
of PDEs posed in unbounded domains. Such methods have a number of important
computational advantages. In particular, they admit very efficient
practical implementation. However, the stability and convergence analysis
of these methods require deep understanding of approximation properties of
the underlying functional basis. In this project, we deal with Chebyshev
spectral and pseudo-spectral methods in unbounded domains. The first part
of the project deals with theoretical analysis of Chebyshev-type spectral projection
and interpolation operators in Bessel potential spaces. In the second
part, we provide rigorous analyses of Chebyshev-type pseudo-spectral (collocation)
scheme applied to the nonlinear Schrodinger equation. The project
is concluded with several numerical experiments.
