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

2017 ◽  
Vol 44 (2) ◽  
pp. 541-570 ◽  
Author(s):  
Sudeep Kundu ◽  
Amiya Kumar Pani
Keyword(s):  
Finite Element ◽  
Feedback Control ◽  
Burgers Equation ◽  
Neumann Boundary ◽  
Finite Element Approximation ◽  
Control Law ◽  
Element Approximation ◽  
Global Stabilization ◽  
Boundary Feedback Control ◽  
Boundary Feedback

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

2020 ◽  
Vol 84 (3) ◽  
Author(s):  
Sudeep Kundu ◽  
Amiya Kumar Pani
Keyword(s):  
Finite Element ◽  
Feedback Control ◽  
Burgers Equation ◽  
Neumann Boundary ◽  
Control Law ◽  
Global Stabilization ◽  
Two Dimensional ◽  
Spatial Direction ◽  
Boundary Feedback Control ◽  
Boundary Feedback

Abstract In this article, global stabilization results for the two dimensional viscous Burgers’ equation, that is, convergence of unsteady solution to its constant steady state solution with any initial data, are established using a nonlinear Neumann boundary feedback control law. Then, applying $$C^0$$ C 0 -conforming finite element method in spatial direction, optimal error estimates in $$L^\infty (L^2)$$ L ∞ ( L 2 ) and in $$L^\infty (H^1)$$ L ∞ ( H 1 ) -norms for the state variable and convergence result for the boundary feedback control law are derived. All the results preserve exponential stabilization property. Finally, several numerical experiments are conducted to confirm our theoretical findings.

An Angular Finite Element Method for the Solution of the Even-Parity Equation

2009 ◽  
Author(s):  
R. Becker ◽  
R. Koch ◽  
M. F. Modest ◽  
H.-J. Bauer
Keyword(s):  
Finite Element ◽  
Finite Element Approximation ◽  
Basis Functions ◽  
New Method ◽  
Discrete Ordinates ◽  
Test Case ◽  
Element Approximation ◽  
Promising Alternative ◽  
Element Discretization ◽  
Spatial Coordinates

The present article introduces a new method to solve the radiative transfer equation (RTE). First, a finite element discretization of the solid angle dependence is derived, wherein the coefficients of the finite element approximation are functions of the spatial coordinates. The angular basis functions are defined according to finite element principles on subdivisions of the octahedron. In a second step, these spatially dependent coefficients are discretized by spatial finite elements. This approach is very attractive, since it provides a concise derivation for approximations of the angular dependence with an arbitrary number of angular nodes. In addition, the usage of high-order angular basis functions is straightforward. In the current paper the governing equations are first derived independently of the actual angular approximation. Then, the design principles for the angular mesh are discussed and the parameterization of the piecewise angular basis functions is derived. In the following, the method is applied to two-dimensional test cases which are commonly used for the validation of approximation methods of the RTE. The results reveal that the proposed method is a promising alternative to the well-established practices like the Discrete Ordinates Method (DOM) and provides highly accurate approximations. A test case known to exhibit the ray effect in the DOM verifies the ability of the new method to avoid ray effects.

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