Non-linear Partition of Unity method

TL;DR

The paper presents a new Non-linear Partition of Unity Method that combines Radial Basis Function interpolation and WENO algorithms to improve high-accuracy approximations near discontinuities, validated through theoretical error bounds and numerical experiments.

Contribution

It introduces a novel non-linear partition of unity approach that adaptively handles discontinuities, enhancing interpolation accuracy over existing methods.

Findings

Improved interpolation accuracy near discontinuities.

Validated error bounds through theoretical analysis.

Numerical experiments confirm theoretical improvements.

Abstract

This paper introduces the Non-linear Partition of Unity Method, a novel technique integrating Radial Basis Function interpolation and Weighted Essentially Non-Oscillatory algorithms. It addresses challenges in high-accuracy approximations, particularly near discontinuities, by adapting weights dynamically. The method is rooted in the Partition of Unity framework, enabling efficient decomposition of large datasets into subproblems while maintaining accuracy. Smoothness indicators and compactly supported functions ensure precision in regions with discontinuities. Error bounds are calculated and validate its effectiveness, showing improved interpolation in discontinuous and smooth regions. Some numerical experiments are performed to check the theoretical results.