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# Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr

IMPACT FACTOR 2017: 0.658

CiteScore 2017: 1.05

SCImago Journal Rank (SJR) 2017: 1.291
Source Normalized Impact per Paper (SNIP) 2017: 0.893

Mathematical Citation Quotient (MCQ) 2017: 0.76

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1609-9389
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Volume 16, Issue 4

# A Nonconforming Finite Element Approximation for Optimal Control of an Obstacle Problem

Asha K. Dond
/ Thirupathi Gudi
/ Neela Nataraj
Published Online: 2016-09-14 | DOI: https://doi.org/10.1515/cmam-2016-0024

## Abstract

The article deals with the analysis of a nonconforming finite element method for the discretization of optimization problems governed by variational inequalities. The state and adjoint variables are discretized using Crouzeix–Raviart nonconforming finite elements, and the control is discretized using a variational discretization approach. Error estimates have been established for the state and control variables. The results of numerical experiments are presented.

MSC 2010: 35J86; 49M25; 65K15; 65N30

## References

• [1]

Braess D., Finite Elements, Theory, Fast Solvers, and Applications in Solid Mechanics, Cambridge University Press, Cambridge, 1997. Google Scholar

• [2]

Brenner S. C. and Scott L. R., The Mathematical Theory of Finite Element Methods, Springer, New York, 1994. Google Scholar

• [3]

Brezzi F., Hager W. W. and Raviart P. A., Error estimates for the finite element solution of variational inequalities. Part I: Primal theory, Numer. Math. 28 (1977), 431–443. Google Scholar

• [4]

Carstensen C. and Köhler K., Non-conforming FEM for the obstacle problem, IMA J Numer Anal. (2016), 10.1093/imanum/drw005. Google Scholar

• [5]

Casas E. and Tröltzsch F., Error estimates for the finite element approximation of a semi-linear elliptic control problem, Control Cybern. 31 (2002), 695–712. Google Scholar

• [6]

Falk R. S., Error estimates for the approximation of a class of variational inequalities, Math. Comp. 28 (1974), 963–971. Google Scholar

• [7]

Fieldler M., Special Matrices and Their Applications in Numerical Mathematics, Martinus Nijhoff, Dordrecht, 1986. Google Scholar

• [8]

Gastaldi L. and Nochetto R., Optimal ${L}^{\mathrm{\infty }}$-error estimates for nonconforming and mixed finite element methods of lowest order, Numer. Math. 50 (1987), 587–611. Google Scholar

• [9]

Glowinski R., Numerical Methods for Nonlinear Variational Problems, Springer, Berlin, 2008. Google Scholar

• [10]

Haslinger J. and Roubíček T., Optimal control of variational inequalities. Approximation theory and numerical realization, Appl. Math. Optim. 14 (1986), 187–201. Google Scholar

• [11]

Herzog R., Rösch A., Ulbrich S. and Wollner W., OPTPDE — A collection of problems in PDE-constrained optimization, Trends in PDE Constrained Optimization, Internat. Ser. Numer. Math. 165, Springer, Cham (2014), 539–543. Google Scholar

• [12]

Hintermüller M., Ito K. and Kunisch K., The primal-dual active set strategy as a semismooth Newton method, SIAM J. Optim. 13 (2002), 865–888. Google Scholar

• [13]

Hintermüller M. and Kopacka I., Mathematical programs with complementarity constraints in function space: C- and strong stationarity and a path-following algorithm, SIAM J. Optim. 20 (2009), 868–902. Google Scholar

• [14]

Kinderlehrer D. and Stampacchia G., An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, 1980. Google Scholar

• [15]

Kunisch K. and Wachsmuth D., Sufficient optimality conditions and semi-smooth Newton methods for optimal control of stationary variational inequalities, ESAIM Control Optim. Calc. Var. 18 (2012), 520–547. Google Scholar

• [16]

Meyer C. and Thoma O., A priori finite element error analysis for optimal control of the obstacle problem, SIAM J. Numer. Anal. 51 (2013), no. 1, 605–628. Google Scholar

• [17]

Mignot F. and Puel J. P., Optimal control in some variational inequalities, SIAM J. Control Optim. 22 (1984), no. 3, 466–476. Google Scholar

• [18]

Nitsche J., ${L}^{\mathrm{\infty }}$ convergence of finite element approximations, Mathematical Aspects of Finite Element Methods, Lecture Notes in Math. 606, Springer, Berlin (1977), 261–274. Google Scholar

• [19]

Scheel H. and Scholtes S., Mathematical programs with complementarity constraints: Stationarity, optimality and sensitivity, Math. Oper. Res. 25 (2000), 1–22. Google Scholar

• [20]

Wang L.-H., On the error estimate of nonconforming finite element approximation to the obstacle problem, J. Comput. Math. 21 (2003), no. 4, 481–490. Google Scholar

Revised: 2016-04-29

Accepted: 2016-08-03

Published Online: 2016-09-14

Published in Print: 2016-10-01

The third author was partially supported by the Simons Foundation and the Oberwolfach Research Institute for Mathematics.

Citation Information: Computational Methods in Applied Mathematics, Volume 16, Issue 4, Pages 653–666, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840,

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