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

TL;DR

This work develops a $C^0$-conforming finite element framework for the Chafee-Infante equation with distributed finite-parameter feedback control, establishing stability and optimal-order error estimates for both state and control in the semi-discrete setting and for a fully discrete scheme with backward Euler time stepping. It provides rigorous existence/uniqueness results for the fully discrete solution and quantifies spatial and temporal convergence respectively, corroborated by numerical experiments that confirm stabilization to the steady state and the predicted convergence rates. The analysis advances understanding of finite-parameter control in nonlinear dissipative PDEs and offers practically relevant guidance for implementing stabilized CI simulations with reliable error bounds.

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.

Figures (3)

• Figure 1: Example 5.1: (i) State variables in the $L^{2}$-norm for both mixed and Neumann boundaries with $\gamma = 9$, $\mu = 20$, and $\nu = 0.1$. (ii) Solution of the state variable in the $L^{2}$-norm for different values of $\gamma$, with fixed $\nu = 0.1, \delta=9,$ and $\mu = 20$. (iii) Control input for both mixed and Neumann boundaries. (iv) Control input for various values of $\gamma$. (v) Uncontrolled solution with $\gamma = 9$, $\mu = 20$, and $\nu = 0.1$. (vi) Controlled solution with $\gamma = 9$, $\mu = 20$, and $\nu = 0.1$.
• Figure 2: Example 5.2: (i) and (ii) Uncontrolled and controlled solutions for different values of $\mu$ with fixed parameters $\nu = 0.5$ and $\gamma = \delta = 1$, (iii) and (iv) Control input for different values of $\mu$ with fixed parameters $\nu = 0.5$ and $\gamma = \delta = 1$, (v) State variable in the $L^{2}-$norm for different values of $\nu$ with fixed $\gamma = 50$ and $\mu = 120$, (vi) State variable in the $L^{2}-$norm for different values of $\delta$ with fixed parameters $\gamma = 50$, $\nu = 0.5$, and $\mu = 120$.
• Figure 3: Example 5.3: (i) Interpolant operators such as Fourier modes, nodal values, and finite volume elements in the $L^{2}-$norm, (ii) Controller for different values of finite parameter feedback control in the $L^{2}-$norm.

Theorems & Definitions (13)

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