Accessible Requires Authentication Published by De Gruyter September 14, 2016

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

Asha K. Dond, Thirupathi Gudi and Neela Nataraj

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.

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

Acknowledgements

A part of the work was completed during the research stay of the second and third authors in the Oberwolfach Research Institute for Mathematics.

References

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

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

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

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

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

[8] Gastaldi L. and Nochetto R., Optimal L-error estimates for nonconforming and mixed finite element methods of lowest order, Numer. Math. 50 (1987), 587–611. Search in Google Scholar

[9] Glowinski R., Numerical Methods for Nonlinear Variational Problems, Springer, Berlin, 2008. Search in 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. Search in 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. Search in 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. Search in 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. Search in Google Scholar

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

[18] Nitsche J., L convergence of finite element approximations, Mathematical Aspects of Finite Element Methods, Lecture Notes in Math. 606, Springer, Berlin (1977), 261–274. Search in Google Scholar

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

Received: 2015-10-5
Revised: 2016-4-29
Accepted: 2016-8-3
Published Online: 2016-9-14
Published in Print: 2016-10-1

© 2016 by De Gruyter