Global properties of pseudospectral methods (Q1118370)

The accuracy of a special pseudospectral algorithm both for approximating functions and numerical solutions of hyperbolic and elliptic differential equations is considered. The derivative matrix for a general sequence of collocation points is explicitly constructed and the authors explore the effect of several factors on the performance of these methods. The result of this consideration is that global methods cannot be interpreted in terms of local methods. For example, the accuracy of the approximation differs when large gradients of the function occur near the center of the region or in the vicinity of the boundary. This difference does not depend on the density of the collocation points near the boundaries. Hence, intuition based on finite difference methods can lead to false results.