Finite Element Analysis for the Chafee-Infante Equation Using Distributed Feedback Control

TL;DR

This paper develops a finite element method with feedback control for the Chafee-Infante equation, providing error analysis, stability, and numerical verification for the fully discrete scheme.

Contribution

It introduces a C^0-conforming finite element approach with finite-parameter feedback control and offers comprehensive error and stability analysis for the discretized system.

Findings

Error estimates for state and control variables

Stability analysis of the fully discrete scheme

Numerical experiments confirming theoretical results

Abstract

In this paper, we propose a \( C^0 \)-conforming finite element method for the Chafee-Infante equation with a finite-parameter feedback control. We establish error analysis for both the state variable and the control variable for the spatially discretized solution. Furthermore, we employ the backward Euler method for time discretization and discuss the stability analysis of the fully discrete scheme. Additionally, we develop error estimates for both the state variable and the control input in the fully discrete setting. Finally, we verify our theoretical conclusions using some numerical experiments.