scholarly journals Numerical approximation of control problems of non-monotone and non-coercive semilinear elliptic equations

2021 ◽  
Author(s):  
Eduardo Casas ◽  
Mariano Mateos ◽  
Arnd Rösch
Keyword(s):  
Control Problem ◽  
Error Estimates ◽  
Numerical Approximation ◽  
Discrete Equation ◽  
Element Approximation ◽  
Continuous Control ◽  
Optimal Controls ◽  
Discrete State ◽  
Semilinear Elliptic ◽  
Continuous Problem

AbstractWe analyze the numerical approximation of a control problem governed by a non-monotone and non-coercive semilinear elliptic equation. The lack of monotonicity and coercivity is due to the presence of a convection term. First, we study the finite element approximation of the partial differential equation. While we can prove existence of a solution for the discrete equation when the discretization parameter is small enough, the uniqueness is an open problem for us if the nonlinearity is not globally Lipschitz. Nevertheless, we prove the existence and uniqueness of a sequence of solutions bounded in $$L^\infty (\varOmega )$$ L ∞ ( Ω ) and converging to the solution of the continuous problem. Error estimates for these solutions are obtained. Next, we discretize the control problem. Existence of discrete optimal controls is proved, as well as their convergence to solutions of the continuous problem. The analysis of error estimates is quite involved due to the possible non-uniqueness of the discrete state for a given control. To overcome this difficulty we define an appropriate discrete control-to-state mapping in a neighbourhood of a strict solution of the continuous control problem. This allows us to introduce a reduced functional and obtain first order optimality conditions as well as error estimates. Some numerical experiments are included to illustrate the theoretical results.

scholarly journals Error estimates of finite volume method for Stokes optimal control problem

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Lin Lan ◽  
Ri-hui Chen ◽  
Xiao-dong Wang ◽  
Chen-xia Ma ◽  
Hao-nan Fu
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Error Estimates ◽  
Finite Volume ◽  
Stokes Equations ◽  
A Priori ◽  
The State ◽  
Element Approximation ◽  
Norm Error

AbstractIn this paper, we discuss a priori error estimates for the finite volume element approximation of optimal control problem governed by Stokes equations. Under some reasonable assumptions, we obtain optimal $L^{2}$ L 2 -norm error estimates. The approximate orders for the state, costate, and control variables are $O(h^{2})$ O ( h 2 ) in the sense of $L^{2}$ L 2 -norm. Furthermore, we derive $H^{1}$ H 1 -norm error estimates for the state and costate variables. Finally, we give some conclusions and future works.

Error estimates for the numerical approximation of a distributed optimal control problem governed by the von Kármán equations

2018 ◽  
Vol 52 (3) ◽  
pp. 1137-1172
Author(s):  
Gouranga Mallik ◽  
Neela Nataraj ◽  
Jean-Pierre Raymond
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Error Estimates ◽  
Numerical Approximation ◽  
The State ◽  
Distributed Optimal Control ◽  
Von Karman ◽  
Von Kármán Equations ◽  
Von Kármán

In this paper, we discuss the numerical approximation of a distributed optimal control problem governed by the von Kármán equations, defined in polygonal domains with point-wise control constraints. Conforming finite elements are employed to discretize the state and adjoint variables. The control is discretized using piece-wise constant approximations. A priori error estimates are derived for the state, adjoint and control variables. Numerical results that justify the theoretical results are presented.

scholarly journals Topological Derivatives for Semilinear Elliptic Equations

2009 ◽  
Vol 19 (2) ◽  
pp. 191-205 ◽  
Author(s):  
Mohamed Iguernane ◽  
Serguei Nazarov ◽  
Jean-Rodolphe Roche ◽  
Jan Sokolowski ◽  
Katarzyna Szulc
Keyword(s):  
Finite Element ◽  
Convergence Rate ◽  
Error Estimates ◽  
Elliptic Equations ◽  
Numerical Experiments ◽  
Finite Element Approximation ◽  
Semilinear Elliptic Equations ◽  
Element Approximation ◽  
Semilinear Elliptic ◽  
Topological Derivatives

Topological Derivatives for Semilinear Elliptic EquationsThe form of topological derivatives for an integral shape functional is derived for a class of semilinear elliptic equations. The convergence of finite element approximation for the topological derivatives is shown and the error estimates in theL∞norm are obtained. The results of numerical experiments which confirm the theoretical convergence rate are presented.

scholarly journals A posteriori error estimates of hp spectral element method for parabolic optimal control problems

2022 ◽  
Vol 7 (4) ◽  
pp. 5220-5240
Author(s):  
Zuliang Lu ◽  
◽  
Fei Cai ◽  
Ruixiang Xu ◽  
Chunjuan Hou ◽  
...  
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Error Estimates ◽  
A Posteriori Error Estimates ◽  
Control Variable ◽  
Posteriori Error ◽  
Spectral Element ◽  
Element Approximation ◽  
Parabolic Optimal Control

<abstract><p>In this paper, we investigate the spectral element approximation for the optimal control problem of parabolic equation, and present a hp spectral element approximation scheme for the parabolic optimal control problem. For improve the accuracy of the algorithm and construct an adaptive finite element approximation. Under the Scott-Zhang type quasi-interpolation operator, a $ L^2(H^1)-L^2(L^2) $ posteriori error estimates of the hp spectral element approximated solutions for both the state variables and the control variable are obtained. Adopting two auxiliary equations and stability results, a $ L^2(L^2)-L^2(L^2) $ posteriori error estimates are derived for the hp spectral element approximation of optimal parabolic control problem.</p></abstract>

scholarly journals L∞-error estimates for a class of semilinear elliptic variational inequalities and quasi-variational inequalities

2001 ◽  
Vol 27 (5) ◽  
pp. 309-319
Author(s):  
M. Boulbrachene ◽  
P. Cortey-Dumont ◽  
J. C. Miellou
Keyword(s):  
Variational Inequalities ◽  
Error Estimates ◽  
Finite Element Approximation ◽  
Banach Fixed Point Theorem ◽  
Element Approximation ◽  
Semilinear Elliptic ◽  
Elliptic Variational Inequalities ◽  
Noncoercive Problems ◽  
The Right ◽  
Uniform Error Estimates

This paper deals with the finite element approximation of a class of variational inequalities (VI) and quasi-variational inequalities (QVI) with the right-hand side depending upon the solution. We prove that the approximation is optimally accurate inL∞combining the Banach fixed point theorem with the standard uniform error estimates in linear VIs and QVIs. We also prove that this approach extends successfully to the corresponding noncoercive problems.

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