Fast multilevel augmentation method for nonlinear integral equations

Abstract

In this study, we extend the multilevel augmentation method for Hammerstein equations established in Chen et al. [Fast multilevel augmentation methods for solving Hammerstein equations, SIAM J. Numer. Anal. 47 (2009), pp. 2321–2346] to solve nonlinear Urysohn integral equations. Under certain differentiability assumptions on the kernel function, we show that the method enjoys the optimal convergence order and linear computational complexity. Finally, numerical experiments are presented to confirm the theoretical results and illustrate the efficiency of the method.

Keywords:

• multilevel augmentation method
• multiscale decomposition
• Urysohn integral equations
• orthogonal projection
• Galerkin scheme

2000 AMS Subject Classifications :

• 65J15
• 65R20

Acknowledgements

The author is very appreciative of the reviewers’ positive and constructive comments and useful suggestions to improve this study.