A two-grid finite element method for nonlinear parabolic integro-differential equations

ABSTRACT

In this paper, we present a two-grid finite element method (FEM) for a two-dimensional nonlinear parabolic integro-differential equation. We solve a fully nonlinear system on a coarse grid space with a grid size H and derive a rough approximation of the exact solution, and then solve the corresponding linearized problem on a fine grid space with a grid size h. The optimal error estimates in $H^1$-norm are obtained for spatially the semidiscrete two-grid FEM. Finally, numerical examples are given to testify the efficiency of the method.

Keywords:

• Two-grid
• nonlinear parabolic integro-differential equations
• finite element method
• error estimates

2010 Mathematics Subject Classifications:

• 65M60
• 65N15
• 65N30

Acknowledgments

The authors thank the referees for valuable constructive comments and suggestions which lead to a significant improvement of this paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was supported by the National Natural Science Foundation of China (11771375, 11571297 and 11601468) and by the Shandong Province Natural Science Foundation (ZR2018QA003 and ZR2018MAQ008).