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BIT Numerical Mathematics

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01-03-2015

Adaptive edge element approximation of H(curl)-elliptic optimal control problems with control constraints

Authors: Ronald H. W. Hoppe, Irwin Yousept

Published in: BIT Numerical Mathematics | Issue 1/2015

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Abstract

A three-dimensional H(curl)-elliptic optimal control problem with distributed control and pointwise constraints on the control is considered. We present a residual-type a posteriori error analysis with respect to a curl-conforming edge element approximation of the optimal control problem. Here, the lowest order edge elements of Nédélec’s first family are used for the discretization of the state and the control with respect to an adaptively generated family of simplicial triangulations of the computational domain. In particular, the a posteriori error estimator consists of element and face residuals associated with the state equation and the adjoint state equation. The main results are the reliability of the estimator and its efficiency up to oscillations in terms of the data of the problem. In the last part of the paper, numerical results are included which illustrate the performance of the adaptive approach.

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Ainsworth, M., Oden, T.: A Posteriori Error Estimation in Finite Element Analysis. Wiley, Chichester (2000)CrossRefMATH

Arnold, D., Falk, R., Winther, R.: Multigrid in H(div) and H(curl). Numer. Math. 85, 197–217 (2000)

Arnold, D., Falk, R., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numerica 15(2006), 1–155 (2006)CrossRefMATHMathSciNet

Babuska, I., Strouboulis, T.: The Finite Element Method and its Reliability. Clarendon Press, Oxford (2001)

Bangerth, W., Rannacher, R.: Adaptive Finite Element Methods for Differential Equations. Lectures in Mathematics. ETH-Zürich. Birkhäuser, Basel (2003)

Beck, R., Deuflhard, P., Hiptmair, R., Hoppe, R.H.W., Wohlmuth, B.: Adaptive multilevel methods for edge element discretizations of Maxwell’s equations. Surv. Math. Ind. 8, 271–312 (1999)

Beck, R., Hiptmair, R., Hoppe, R.H.W., Wohlmuth, B.: Residual based a posteriori error estimators for eddy current computation. M2AN Math. Model. Numer. Anal. 34, 159–182 (2000)CrossRefMATHMathSciNet

Binev, P., Dahmen, W., DeVore, R.: Adaptive finite element methods with convergence rates. Numer. Math. 97, 219–268 (2004)CrossRefMATHMathSciNet

Buffa, A., Ciarlet Jr, P.: On traces for functional spaces related to Maxwell’s equations. part i. Math. Meths. Appl. Sci. 24, 9–30 (2001)

10.

Carstensen, C., Hoppe, R.H.W.: Convergence analysis of an adaptive edge finite element method for the 2d eddy current equations. J. Numer. Math. 13, 19–32 (2005)CrossRefMATHMathSciNet

11.

Carstensen, C., Hoppe, R.H.W.: Unified framework for an a posteriori error analysis of non-standard finite element approximations of H(curl)-elliptic problems. J. Numer. Math. 17, 27–44 (2009)CrossRefMATHMathSciNet

12.

Cascon, J.M., Kreuzer, Ch., Nochetto, R.H., Siebert, K.G.: Quasi-optimal rate of convergence of adaptive finite element methods. SIAM J. Numer. Anal. 46, 2524–2550 (2008)CrossRefMATHMathSciNet

13.

Costabel, M., Dauge, M.: Singularities of electromagnetic fields in polyhedral domains. Arch. Ration. Mech. Anal. 151, 221–276 (2000)CrossRefMATHMathSciNet

14.

Dörfler, W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33, 1106–1124 (1996)

15.

Eriksson, K., Estep, D., Hansbo, P., Johnson, C.: Computational Differential Equations. Cambridge University Press, Cambridge (1996)MATH

16.

Gaevskaya, A., Hoppe, R.H.W., Iliash, Y., Kieweg, M.: Convergence analysis of an adaptive finite element method for distributed control problems with control constraints. In: Leugering, G., et al., (eds.) Proc. Conf. Optimal Control for PDEs, Oberwolfach, Germany, Birkhäuser, Basel (2007)

17.

Hintermüller, M., Hoppe, R.H.W.: Goal-oriented adaptivity in control constrained optimal control of partial differential equations. SIAM J. Control Optim. 47, 1721–1743 (2008)CrossRefMATHMathSciNet

18.

Hintermüller, M., Hoppe, R.H.W., Iliash, Y., Kieweg, M.: An a posteriori error analysis of adaptive finite element methods for distributed elliptic control problems with control constraints. ESAIM Control Optim. Calc. Var. 14, 540–560 (2008)CrossRefMATHMathSciNet

19.

Hiptmair, R.: Multigrid method for Maxwell’s equations. SIAM J. Numer. Anal. 36, 204–225 (1999)

20.

Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numerica 11, 237–339 (2002)CrossRefMATHMathSciNet

21.

Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms. Springer, Berlin-Heidelberg-New York (1993)

22.

Hoppe, R.H.W., Schöberl, J.: Convergence of adaptive edge element methods for the 3d eddy currents equations. J. Comp. Math. 27, 657–676 (2009)CrossRefMATH

23.

Houston, P., Perugia, I., Schötzau, D.: A posteriori error estimation for discontinuous Galerkin discretizations of H(curl)-elliptic partial differential equations. IMA J. Numer. Anal. 27, 122–150 (2007)

24.

Kolmbauer, M., Langer, U.: A robust preconditioned minres solver for distributed time-periodic eddy current optimal control problems. SIAM J. Sci. Comput. 34, B785–B809 (2012)CrossRefMATHMathSciNet

25.

Li, R., Liu, W., Ma, H., Tang, T.: Adaptive finite element approximation for distributed elliptic optimal control problems. SIAM J. Control Optim. 41, 1321–1349 (2002)CrossRefMATHMathSciNet

26.

Liu, W., Yan, N.: Adaptive Finite Element Methods for Optimal Control Governed by PDEs. Series in Information and Computational Science, vol. 41. Global Science Press, Hong Kong (2008)

27.

Logg, A., Mardal, K.-A., Wells, G.N.: Automated Solution of Differential Equations by the Finite Element Method. Springer, Boston (2012)CrossRefMATH

28.

Monk, P.: A posteriori error indicators for Maxwell’s equations. J. Comp. Appl. Math. 100, 173–190 (1998)

29.

Monk, P.: Finite Element Methods for Maxwell Equations. Oxford University Press, Oxford (2003)CrossRefMATH

30.

Nédélec, J.-C.: Mixed finite elements in \(\mathbb{R}^{3}\). Numer. Math. 35, 315–341 (1980)

31.

Neittaanmäki, P., Repin, S.: Reliable Methods for Mathematical Modelling. Error Control and a Posteriori Estimates. Elsevier, New York (2004)

32.

Schöberl, J.: A posteriori error estimates for Maxwell equations. Math. Comp. 77, 633–649 (2008)

33.

Stevenson, R.: Optimality of a standard adaptive finite element method. Found. Comput. Math. 7, 245–269 (2007)CrossRefMATHMathSciNet

34.

Tartar, L.: Introduction to Sobolev Spaces and Interpolation Theory. Springer, Berlin (2007)

35.

Tröltzsch, F.: Optimal Control of Partial Differential Equations. Theory, Methods, and Applications. American Mathematical Society, Providence (2010)

36.

Tröltzsch, F., Yousept, I.: PDE-constrained optimization of time-dependent 3D electromagnetic induction heating by alternating voltages. ESAIM M2AN 46, 709–729 (2012)CrossRefMATH

37.

Verfürth, R.: A Review of a Posteriori Estimation and Adaptive Mesh—Refinement Techniques. Wiley-Teubner, New York (1996)MATH

38.

Vexler, B., Wollner, W.: Adaptive finite elements for elliptic optimization problems with control constraints. SIAM J. Control Optim. 47, 1150–1177 (2008)CrossRefMATHMathSciNet

39.

Zhong, L., Chen, L., Shu, S., Wittum, G., Xu, J.: Optimal error estimates of the Nedelec edge elements for time-harmonic Maxwell’s equations. J. Comput. Math. 27(2009), 563–572 (2009)

40.

Yousept, I.: Optimal control of quasilinear H (curl)-elliptic partial differential equations in magnetostatic field problems. SIAM J. Control Optim. 51, 3624–3651 (2013)CrossRefMATHMathSciNet

41.

Yousept, I.: Optimal control of Maxwell’s equations with regularized state constraints. Comput. Optim. Appl. 52, 59–581 (2012)

42.

Yousept, I.: Finite element analysis of an optimal control problem in the coefficients of time-harmonic eddy current equations. J. Optim. Theory Appl. 154, 879–903 (2012)CrossRefMATHMathSciNet

43.

Zhong, L., Chen, L., Shu, S., Wittum, G., Xu, J.: Convergence and optimality of adaptive edge finite element methods for time-harmonic Maxwell equations. Math. Comp. 81, 623–642 (2012)

Title: Adaptive edge element approximation of H(curl)-elliptic optimal control problems with control constraints
Authors: Ronald H. W. Hoppe
Irwin Yousept
Publication date: 01-03-2015
Publisher: Springer Netherlands
Published in: BIT Numerical Mathematics / Issue 1/2015
Print ISSN: 0006-3835
Electronic ISSN: 1572-9125
DOI: https://doi.org/10.1007/s10543-014-0497-x

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