Meshless local Petrov Galerkin method for 2D/3D nonlinear convection–diffusion equations based on LS-RBF-PUM

Abstract

In this article, we propose a meshless local Petrov Galerkin (MLPG) method based on least square radial basis function partition of unity method (LS-RBF-PUM), which is applied to the nonlinear convection–diffusion equations. The proposed method is not sensitive to the node layout, and has good stability and flexibility to complex domain. In order to treat nonlinear term, Picard iterative scheme is employed to confirm the convergence of iterative process. Error estimates are derived by the radial basis function interpolation method and convergence rate is proven to be second order. Numerical examples are performed for the nonlinear convection–diffusion equations in two and three space dimensions (2D/3D), which not only supports the theoretical results but also finds out superconvergence of third order.

Acknowledgments

The authors would like to thank the editor and referees for their valuable comments and suggestions which helped us to improve the results of this article.

Additional information

Funding

This work is in parts supported by the Excellent Doctor Innovation Program of Xinjiang University (No. XJUBSCX-2017006), the Natural Science Foundation of Xinjiang Province (No.2016D01C071) and the NSF of China (No. 11701196, No. 11671345, No. 11471047).