Two-grid economical algorithms for parabolic integro-differential equations with nonlinear memory

TL;DR

This paper introduces efficient two-grid finite element algorithms for solving parabolic integro-differential equations with nonlinear memory, achieving comparable accuracy to standard methods while reducing computational costs.

Contribution

The paper develops and analyzes two-grid algorithms for PIDEs with nonlinear memory, demonstrating stability and efficiency improvements over traditional methods.

Findings

Algorithms are as stable as standard methods.

Achieve same accuracy with reduced computational cost.

Numerical experiments confirm theoretical results.

Abstract

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.