Finite element approximation to global stabilization of the Burgers’ equation by Neumann boundary feedback control law

Global Stabilization of Two Dimensional Viscous Burgers’ Equation by Nonlinear Neumann Boundary Feedback Control and Its Finite Element Analysis

“…Without loss of generality, we assume that w d ≥ 0. When ν is sufficiently small and initial condition u 0 is antisymmetric, the numerical solution of (1.1)-(1.3) with ∂u ∂n = 0 may converge to a nonconstant steady state solution for which related references are given in [19]. We do not consider such cases here.…”

“…Also in [5,6], authors have considered finite element method to solve numerically local stabilization problem for 1D Burgers' equation without any convergence analysis. Subsequently in [19], optimal error estimates in the context of finite element method for the state variable and superconvergence result for the feedback control laws are derived. For related analysis on Benjamin Bona Mahony Burgers' (BBM-Burgers') type equations, we refer to [20].…”

“…For related analysis on Benjamin Bona Mahony Burgers' (BBM-Burgers') type equations, we refer to [20]. Concerning extensive literature for one dimensional Burgers' and BBM-Burgers' problem, see the references in [19,20].…”

Global Stabilization of BBM–Burgers’ Type Equations by Nonlinear Boundary Feedback Control Laws: Theory and Finite Element Error Analysis

“…For related issues of finite element analysis of the viscous Burgers' equation using nonlinear Neumann boundary feedback control law, we refer to our recent article [16]. Compared to [16], special care has been taken to establish global stabilization results in L ∞ (H i )(i = 0, 1, 2) norms as µ → 0. It is further observed that the decay rate for the BBM-B type equation is less than the decay rate for the viscous Burgers' equation and as the dispersion coefficient µ approaches zero, the decay rate also converges to the decay rate for the Burgers' equation.…”

Global Stabilization of Two Dimensional Viscous Burgers' Equation by Nonlinear Neumann Boundary Feedback Control and its Finite Element Analysis

“…Although, these control laws are computed using finite element methods, but convergence of numerical solution posses some serious difficulty because of the typical nonlinearity present in the system through nonlinear feedback laws. Subsequently in [14], optimal error estimates in the context of finite element method for the state variable and superconvergence result for the feedback control laws are derived. For related analysis on BBM-Burgers' type equation, we refer to [15].…”