Basis Functions for Scattered Data Quasi-Interpolation

Abstract

Given scattered data of a smooth function in ${I\!R}^d$, we consider quasi-interpolation operators for approximating the function. In order to use these operators for the derivation of useful schemes for PDE solvers, we would like the quasi-interpolation operators to be of compact support and of high approximation power. The quasi-interpolation operators are generated through known quasi-interpolation operators on uniform grids, and the resulting basis functions are represented by finite combinations of box-splines. A special attention is given to point-sets of varying density. We construct basis functions with support sizes and approximation power related to the local density of the data points. These basis functions can be used in Finite Elements and in Isogeometric Analysis in cases where a non-uniform mesh is required, the same as T-splines are being used as basis functions for introducing local refinements in flow problems.