Non-linear Partition of Unity method

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.