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

TL;DR

This paper introduces a fully discrete finite element scheme with theta time discretization for the viscous Burgers' equation with nonlinear Neumann boundary control, proving stability, convergence, and validating through numerical experiments.

Contribution

It develops a novel unconditionally stable finite element theta scheme for Burgers' equations with boundary control, providing rigorous error estimates and numerical validation.

Findings

Unconditional exponential stability for the scheme.

Optimal error estimates for state and control.

Numerical experiments confirming theoretical results.

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 $\theta$-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 $\theta \in [1/2, 1]$, 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.