Finite element theta schemes for the viscous Burgers' equation with nonlinear Neumann boundary feedback control

TL;DR

This work addresses stabilization and numerical approximation of the viscous Burgers' equation with nonlinear Neumann boundary feedback control in both 1D and 2D. It develops a fully discrete scheme based on a $\theta$-time discretization and a $C^{0}$-conforming finite element spatial discretization, proving unconditional exponential stability for $\theta\in[\tfrac{1}{2},1]$ and providing optimal error estimates for state and boundary control. Existence and uniqueness results accompany the stability and error analyses in both dimensions, with extensive numerical experiments validating the theoretical findings and demonstrating practical stabilization effects. Overall, the approach preserves the continuous stabilization properties at the discrete level and delivers reliable convergence behavior across dimensions.

Abstract

In this article, we develop a fully discrete numerical scheme for the one-dimensional (1D) and two-dimensional (2D) viscous Burgers equations with nonlinear Neumann boundary feedback control. The temporal discretization employs a $θ$-scheme, while a conforming finite element method is used for the spatial approximation. The existence and uniqueness of the fully discrete solution are established. We further prove that the scheme is unconditionally exponentially stable for $θ\in [1/2, 1]$, thereby ensuring that the stabilization property of the continuous model is retained at the discrete level. In addition, optimal error estimates are obtained for both the state variable and the boundary control inputs in 1D and 2D frameworks. Finally, several numerical experiments are presented to validate our theoretical findings and to demonstrate the effectiveness of the proposed stabilization strategy under varying model parameters.

Figures (3)

• Figure 1: Example \ref{['ex1']}: (i) Controlled and uncontrolled solution in the $L^{2}$-norm. (ii) Controlled solution in the $L^{2}-$norm for various values of $\nu$. (iii) In semi-log scale, controlled solution in the $L^{2}-$norm for different values of $\theta$. (iv) Control input for numerous values of $c_{0} \ \text{and} \ c_{1}$ at the left boundary $x=0$. (v) Control input for various values of $c_{0}\ \text{and} \ c_{1}$ at the right boundary $x=1$.
• Figure 2: Example \ref{['ex1']}: (i) Control input for numerous values of $\nu$ at the left boundary $x=0$. (ii) Control input for different values of $\nu$ at the right boundary $x=1$. (iii) Control input for various values of $\theta$ at the left boundary $x=0$. (iv) Control input for various values of $\theta$ at the right boundary $x=1$.
• Figure 3: Example \ref{['ex2']}: (i) In semi-log scale, uncontrolled and controlled solution in the $L^{2}-$norm. (ii) In semi-log scale, control input for different values of $c_{2}$ with $\nu=1$ in the $L^{2}$-norm. (iii) Controlled solution in the $L^{2}$-norm for various values of $\nu$ with $c_{2}=0.1$. (iv) In semi-log scale, control input for various values of $\nu$ in the $L^{2}-$norm with $c_{2}=0.1$.

Theorems & Definitions (16)

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