This article discusses the numerical analysis of the distributed optimal control problem governed by the von Kármán equations defined on a polygonal domain in ℝ2. The state and adjoint variables are discretised using the nonconforming Morley finite element method and the control is discretized using piecewise constant functions. A priori and a posteriori error estimates are derived for the state, adjoint and control variables. The a posteriori error estimates are shown to be efficient. Numerical results that confirm the theoretical estimates are presented.
Keywords: von Kármán equations, distributed control, plate bending, non-linear, nonconforming, Morley FEM, $$, $$, error estimates
@article{M2AN_2022__56_5_1655_0,
author = {Chowdhury, Sudipto and Dond, Asha K. and Nataraj, Neela and Shylaja, Devika},
title = {\protect\emph{A posteriori} error analysis for a distributed optimal control problem governed by the von {K\'arm\'an} equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1655--1686},
year = {2022},
publisher = {EDP-Sciences},
volume = {56},
number = {5},
doi = {10.1051/m2an/2022040},
mrnumber = {4454164},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2022040/}
}
TY - JOUR AU - Chowdhury, Sudipto AU - Dond, Asha K. AU - Nataraj, Neela AU - Shylaja, Devika TI - A posteriori error analysis for a distributed optimal control problem governed by the von Kármán equations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2022 SP - 1655 EP - 1686 VL - 56 IS - 5 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2022040/ DO - 10.1051/m2an/2022040 LA - en ID - M2AN_2022__56_5_1655_0 ER -
%0 Journal Article %A Chowdhury, Sudipto %A Dond, Asha K. %A Nataraj, Neela %A Shylaja, Devika %T A posteriori error analysis for a distributed optimal control problem governed by the von Kármán equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2022 %P 1655-1686 %V 56 %N 5 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2022040/ %R 10.1051/m2an/2022040 %G en %F M2AN_2022__56_5_1655_0
Chowdhury, Sudipto; Dond, Asha K.; Nataraj, Neela; Shylaja, Devika. A posteriori error analysis for a distributed optimal control problem governed by the von Kármán equations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 56 (2022) no. 5, pp. 1655-1686. doi: 10.1051/m2an/2022040
[1] , , and , A posteriori error estimates for a distributed optimal control problem of the stationary Navier-Stokes equations. SIAM J. Control Optim. 59 (2021) 2898–2923. | MR | DOI
[2] , On von Kármán equations and the buckling of a thin elastic plate, I the clamped plate. Comm. Pure Appl. Math. 20 (1967) 687–719. | MR | Zbl | DOI
[3] and , On von Kármán equations and the buckling of a thin elastic plate. Bull. Amer. Math. Soc. 72 (1966) 1006–1011. | MR | Zbl | DOI
[4] and , Von Kármán equations and the buckling of a thin elastic plate. II plate with general edge conditions. Comm. Pure Appl. Math. 21 (1968) 227–241. | MR | Zbl | DOI
[5] and , On the boundary value problem of the biharmonic operator on domains with angular corners. Math. Methods Appl. Sci. 2 (1980) 556–581. | MR | Zbl | DOI
[6] and , interior penalty methods for fourth order elliptic boundary value problems on polygonal domains. J. Sci. Comput. 22-23 (2005) 83–118. | MR | Zbl | DOI
[7] , , and , A Morley finite element method for the displacement obstacle problem of clamped Kirchhoff plates. J. Comput. Appl. Math. 254 (2013) 31–42. | MR | Zbl | DOI
[8] , , and , A interior penalty method for a von Kármán plate. Numer. Math. 135 (2017) 803–832. | MR | DOI
[9] , Finite element approximations of the von Kármán equations. RAIRO Anal. Numér. 12 (1978) 303–312. | MR | Zbl | Numdam | DOI
[10] and , Adaptive Morley FEM for the von Kármán equations with optimal convergence rates. SIAM J. Numer. Anal. 59 (2021) 696–719. | MR | DOI
[11] and , A priori and a posteriori error analysis of the Crouzeix-Raviart and Morley FEM with original and modified right-hand sides. Comput. Methods Appl. Math. 21 (2021) 289–315. | MR | DOI
[12] and , How to prove the discrete reliability for nonconforming finite element methods. J. Comput. Math 38 (2020) 142–175. | MR | DOI
[13] , and , A discrete Helmholtz decomposition with Morley finite element functions and the optimality of adaptive finite element schemes. Comput. Math. Appl. 68 (2014) 2167–2181. | MR | DOI
[14] , and , A priori and a posteriori error control of discontinuous Galerkin finite element methods for the von Kármán equations. IMA J. Numer. Anal. 39 (2019) 167–200. | MR
[15] , and , Nonconforming finite element discretization for semilinear problems with trilinear nonlinearity. IMA J. Numer. Anal. 41 (2021) 164–205. | MR | DOI
[16] , and , Error estimates for the numerical approximation of a distributed control problem for the steady-state Navier-Stokes equations. SIAM J. Control Optim. 46 (2007) 952–982 (electronic). | MR | Zbl | DOI
[17] , and , A mixed finite element scheme for biharmonic equation with variable coefficient and von Kármán equations. (2020). | arXiv | MR
[18] , and , A framework for the error analysis of discontinuous finite element methods for elliptic optimal control problems and applications to IP methods. Numer. Funct. Anal. Optim. 36 (2015) 1388–1419. | MR | DOI
[19] , and , Morley FEM for a distributed optimal control problem governed by the von Kármán equations. Comput. Methods Appl. Math. 21 (2021) 233–262. | MR | DOI
[20] , The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). | MR | Zbl
[21] , Mathematical Elasticity: Theory of Plates. Vol. II. North-Holland, Amsterdam (1997). | MR | Zbl
[22] and , Mathematical Aspects of Discontinuous GAlerkin Methods. Vol. 69 of Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer, Heidelberg (2012). | MR | Zbl | DOI
[23] and , Theory and Practice of Finite Elements. Vol. 159 of Applied Mathematical Sciences. Springer-Verlag, New York (2004). | MR | Zbl | DOI
[24] , Morley finite element method for the eigenvalues of the biharmonic operator. IMA J. Numer. Anal. 35 (2015) 1779–1811. | MR | DOI
[25] , Singularities in Boundary Value Problems. Vol. RMA 22. Masson & Springer-Verlag (1992). | MR | Zbl
[26] and , Finite element approximation of optimal control problems for the von Kármán equations. Numer. Methods Part. Differ. Equ. 11 (1995) 111–125. | MR | Zbl | DOI
[27] and , The best norm error estimate of lower order finite element methods for the fourth order problem. J. Comput. Math. 30 (2012) 449–460. | MR | Zbl | DOI
[28] , Approximation methods for nonlinear problems with application to two-point boundary value problems. Math. Comput. 29 (1975) 464–474. | MR | Zbl | DOI
[29] , An existence theorem for the von Kármán equations. Arch. Ration. Mech. Anal. 27 (1967) 233–242. | MR | Zbl | DOI
[30] , Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971). | MR | Zbl | DOI
[31] and , Conforming finite element methods for the von Kármán equations. Adv. Comput. Math. 42 (2016) 1031–1054. | MR | DOI
[32] and , A nonconforming finite element approximation for the von Kármán equations. ESAIM: M2AN 50 (2016) 433–454. | MR | Numdam | Zbl | DOI
[33] , and , Error estimates for the numerical approximation of a distributed optimal control problem governed by the von Kármán equations. ESAIM: M2AN 52 (2018) 1137–1172. | MR | Zbl | Numdam | DOI
[34] , A mixed finite element method for the solution of the von Kármán equations. Numer. Math. 26 (1976) 255–269. | MR | Zbl | DOI
[35] , Optimal Control of Partial Differential Equations: Theory, Methods and Applications. Vol. 112 of Graduate Studies in Mathematics. Translated from the 2005 German original by Jürgen Sprekels. American Mathematical Society, Providence, RI (2010). | MR | Zbl
[36] , A Posteriori Error Estimation Techniques for Finite Element Methods. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2013). | MR | Zbl | DOI
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