ABSTRACT:
Spectral methods are usually based on
polynomial or trigonometric expansions
and are known for their fast convergence (exponential for
smooth problems). Solutions can be
efficiently computed through fast Fourier transforms, making them popular in the
study of turbulent
flows, meteorological simulations, and
imaging. Conventional spectral methods,
however, have limitations which have prevented them from
being used in many applications where
finite differences and finite elements
are predominant. One obstacle is the need of special nodal
sets for accurate approximations. In this talk we will explore three viable alternatives to circumvent this
difficulty: radial basis function,
hybrid,
and underdetermined methods. Of particular interest is how the accuracy and stability of these
schemes depend on node location
and the
geometry of the problem. Adaptive implementations will also be considered.