Collection of abstracts 13th GAMM-Seminar Kiel on Numerical Treatment of Multi-Scale Problems January 24th to 26th, 1997.

Friday, January 24th, 1997

Multi-grid for plate bending in scattered data interpolation

We consider the scattered data interpolation problem. Given a set of function values in a two dimensional domain, one wants to construct a smooth global interpolation function through the data. The interpolant is determined by a curvature minimization criterion, fulfilling the Kirchhoff plate bending or biharmonic equation. This construction is analogue to the curvature minimization property of one-dimensional splines. The approach is motivated by an implementation of the plate bending interpolation method in the commercial package Siscat (Sintef Scattered Data Library).

We construct a multi-grid method for the biharmonic equation. The problem is the correct treatment of the interpolation conditions represented as prescribed function values in single points. Usually the interpolation points are not present on a coarse grid, since the position of the interpolation points is completely random. We construct the coarse grid matrices by means of Galerkin products which implements the necessary modifications in case of prescribed points on the finest grid. The restriction and prolongation is modified accordingly.