Two-grid economical algorithms for parabolic integro-differential equations with nonlinear memory
For researchers solving PIDEs with nonlinear memory, this work offers a more efficient computational method, though it is an incremental improvement over existing two-grid techniques.
This paper presents two-grid finite element algorithms for parabolic integro-differential equations with nonlinear memory, achieving the same accuracy as the standard algorithm while saving significant storage and computing time when the coarse and fine grid sizes satisfy H=O(h^{(r-1)/r}).
In this paper, several two-grid finite element algorithms for solving parabolic integro-differential equations (PIDEs) with nonlinear memory are presented. Analysis of these algorithms is given assuming a fully implicit time discretization. It is shown that these algorithms are as stable as the standard fully discrete finite element algorithm, and can achieve the same accuracy as the standard algorithm if the coarse grid size $H$ and the fine grid size $h$ satisfy $H=O(h^{\frac{r-1}{r}})$. Especially for PIDEs with nonlinear memory defined by a lower order nonlinear operator, our two-grid algorithm can save significant storage and computing time. Numerical experiments are given to confirm the theoretical results.