The two-grid interpolating element free Galerkin (TG-IEFG) method for solving Rosenau-regularized long wave (RRLW) equation with error analysis

ABSTRACT

The two-grid method is a technique to solve the linear system of algebraic equations for reducing the computational cost. In this study, the two-grid procedure has been combined with the EFG method for solving nonlinear partial differential equations. The two-grid FEM has been introduced in various forms. The well-known two-grid FEM is a three-step method that has been proposed by Bajpai and Nataraj (Comput. Math. Appl. 2014;68:2277–2291) that the new proposed scheme is an ecient procedure for solving important nonlinear partial differential equations such as Navier–Stokes equation. By applying shape functions of IMLS approximation in the EFG method, a new technique that is called interpolating EFG (IEFG) can be obtained. In the current investigation, we combine the two-grid algorithm with the IEFG method for solving the nonlinear Rosenau-regularized long-wave (RRLW) equation. In other hand, we demonstrate that solutions of steps 1, 2, and 3 exist and are unique and also we achieve an error estimate for them. Moreover, three test problems in one- and two-dimensional cases are given which support accuracy and efficiency of the proposed scheme.

KEYWORDS:

• Meshless method
• element-free Galerkin method
• two-grid technique
• nonlinear Rosenau-regularized long-wave (RRLW) equation
• error estimate and convergence analysis

AMS SUBJECT CLASSIFICATIONS:

• 65M12
• 65N12
• 65N30
• 65N40

Acknowledgements

We would like to give our sincere gratitude to the reviewers for their comments and suggestions that greatly improved the manuscript.

Notes

No potential conflict of interest was reported by the authors.