A preconditioned implementation of pseudospectral methods on arbitrary grids. (Q1415259)

Pseudospectral methods basing on an expansion of ultra-spherical polynomials are considered to approximate the solutions of boundary value problems for ordinary and partial differential equations. To overcome the bad conditional properties of those methods, a preconditioner that uses the \textit{differentiation matrices} only indirectly is developed. This leads to a method with a condition number of \(O(n)\), where \(n\) describes the degree of the polynomials. Four examples that verify the theoretical results are presented.