Error estimations for some meshless boundary interpolation methods

Abstract

Meshless methods have become quite popular in numerical treatment of partial differential equations because of their simplicity and the fact that they require neither domain nor boundary mesh. In general, however, they convert the original problem to a highly ill-conditioned linear system of algebraic equations with a dense matrix. Recently, a special technique has been proposed which circumvents this computational difficulty. This method, called Direct Multi-Elliptic Interpolation Method, is based on a scattered data interpolation which defines the interpolation function as a solution of a higher order multi-elliptic equation. Here the boundary version of this meshless method which is based on a multi-elliptic boundary interpolation is considered. Error estimations are derived justifying the interpolation function to be a good approximation of the solution of the original boundary value problem as well. At the same time, the problem of large, dense and ill-conditioned matrices as well as the mesh generation are completely avoided. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)