A posteriori error analysis for the optimal control of magneto-static fields
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 6, pp. 2159-2191.

This paper is concerned with the analysis and numerical investigations for the optimal control of first-order magneto-static equations. Necessary and sufficient optimality conditions are established through a rigorous Hilbert space approach. Then, on the basis of the optimality system, we prove functional a posteriori error estimators for the optimal control, the optimal state, and the adjoint state. 3D numerical results illustrating the theoretical findings are presented.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2017008
Classification : 35Q61, 78A25, 78A30, 65N15, 47A05, 35F05, 35F15
Mots clés : Maxwell’s equations, magneto statics, optimal control, a posteriori error analysis
@article{M2AN_2017__51_6_2159_0,
     author = {Pauly, Dirk and Yousept, Irwin},
     title = {A posteriori error analysis for the optimal control of magneto-static fields},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {2159--2191},
     publisher = {EDP-Sciences},
     volume = {51},
     number = {6},
     year = {2017},
     doi = {10.1051/m2an/2017008},
     mrnumber = {3745168},
     zbl = {1383.35223},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2017008/}
}
TY  - JOUR
AU  - Pauly, Dirk
AU  - Yousept, Irwin
TI  - A posteriori error analysis for the optimal control of magneto-static fields
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2017
SP  - 2159
EP  - 2191
VL  - 51
IS  - 6
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an/2017008/
DO  - 10.1051/m2an/2017008
LA  - en
ID  - M2AN_2017__51_6_2159_0
ER  - 
%0 Journal Article
%A Pauly, Dirk
%A Yousept, Irwin
%T A posteriori error analysis for the optimal control of magneto-static fields
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2017
%P 2159-2191
%V 51
%N 6
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an/2017008/
%R 10.1051/m2an/2017008
%G en
%F M2AN_2017__51_6_2159_0
Pauly, Dirk; Yousept, Irwin. A posteriori error analysis for the optimal control of magneto-static fields. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 6, pp. 2159-2191. doi : 10.1051/m2an/2017008. http://archive.numdam.org/articles/10.1051/m2an/2017008/

S. Bauer, D. Pauly and M. Schomburg, The Maxwell compactness property in bounded weak Lipschitz domains with mixed boundary conditions. SIAM J. Math. Anal. 48 (2016) 2912–2943. | DOI | MR | Zbl

M. Bebendorf, A note on the Poincaré inequality for convex domains. Z. Anal. Anwendungen 22 (2003) 751–756. | DOI | MR | Zbl

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

V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms. Series in Computational Mathematics. Springer Heidelberg (1986). | MR | Zbl

R. Hiptmair, Finite elements in computational electromagnetism. Acta Numer. 11 (2002) 237–339. | DOI | MR | Zbl

R.H.W. Hoppe and I. Yousept, Adaptive edge element approximation ofH(curl)-elliptic optimal control problems with control constraints. BIT 55 (2015) 255–277. | DOI | MR

F. Jochmann, A compactness result for vector fields with divergence and curl in L q (Ω) involving mixed boundary conditions. Appl. Anal. 66 (1997) 189–203. | DOI | MR | Zbl

M. Kolmbauer and U. Langer, Efficient solvers for some classes of time-periodic eddy current optimal control problems. In Numerical Solution of Partial Differential Equations: Theory, Algorithms, and Their Applications, edited by Oleg P. Iliev, Svetozar D. Margenov, Peter D Minev, Panayot S. Vassilevski and Ludmil T Zikatanov. Vol. 45, Springer New York (2013). | MR | Zbl

M. Kolmbauer and U. Langer, A robust preconditioned MinRes solver for time-periodic eddy current problems. Comput. Methods Appl. Math. 13 (2013) 1–20. | DOI | MR | Zbl

R. Leis, Initial Boundary Value Problems in Mathematical Physics. Teubner, Stuttgart (1986). | MR | Zbl

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

O. Mali, P. Neittaanmäki and S. Repin, Accuracy verification methods, theory and algorithms. Springer (2014). | MR | Zbl

J.-C. Nédélec, Mixed finite elements in 𝐑 3 . Numer. Math. 35 (1980) 315–341. | DOI | MR | Zbl

P. Neittaanmäki and S. Repin, Reliable methods for computer simulation, error control and a posteriori estimates. Elsevier, New York (2004). | MR | Zbl

S. Nicaise, S. Stingelin and F. Tröltzsch, On two optimal control problems for magnetic fields. Comput. Methods Appl. Math. 14 (2014) 555–573. | DOI | MR | Zbl

S. Nicaise, S. Stingelin and F. Tröltzsch, Optimal control of magnetic fields in flow measurement. Discrete Contin. Dyn. Syst. Ser. S 8 (2015) 579–605. | MR | Zbl

D. Pauly, On constants in Maxwell inequalities for bounded and convex domains. Zapiski POMI 435 (2014) 46–54, J. Math. Sci. (N. Y.) 210 (2015) 787–792. | MR

D. Pauly. On Maxwell’s and Poincaré’s constants. Discrete Contin. Dyn. Syst. Ser. S 8 (2015) 607–618. | MR | Zbl

D. Pauly, On the Maxwell Constants in 3D, Math. Methods Appl. Sci. 40 (2017) 435–447. | DOI | MR | Zbl

D. Pauly and S. Repin, Two-Sided A Posteriori Error Bounds for Electro-Magneto Static Problems. Zapiski POMI 370 (2009) 94–110; J. Math. Sci. (N. Y.) 166 (2010) 53–62. | MR | Zbl

L.E. Payne and H.F. Weinberger, An optimal Poincaré inequality for convex domains. Arch. Ration. Mech. Anal. 5 (1960) 286–292. | DOI | MR | Zbl

R. Picard, An elementary proof for a compact imbedding result in generalized electromagnetic theory. Math. Z. 187 (1984) 151–164. | DOI | MR | Zbl

R. Picard, N. Weck and K.-J. Witsch, Time-harmonic Maxwell equations in the exterior of perfectly conducting, irregular obstacles. Analysis 21 (2001) 231–263. | DOI | MR | Zbl

S. Repin, A posteriori estimates for partial differential equations. Radon Series Comp. Appl. Math. Walter de Gruyter Berlin (2008). | MR | Zbl

F. Tröltzsch and A. Valli, Optimal control of low-frequency electromagnetic fields in multiply connected conductors. Optimization 65 (2016) 1651–1673. | DOI | MR | Zbl

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

C. Weber, A local compactness theorem for Maxwell’s equations. Math. Methods Appl. Sci. 2 (1980) 12–25. | DOI | MR | Zbl

N. Weck, Maxwell’s boundary value problems on Riemannian manifolds with nonsmooth boundaries. J. Math. Anal. Appl. 46 (1974) 410–437. | DOI | MR | Zbl

K.-J. Witsch, A remark on a compactness result in electromagnetic theory. Math. Methods Appl. Sci. 16 (1993) 123–129. | DOI | MR | Zbl

Y. Xu and J. Zou, A convergent adaptive edge element method for an optimal control problem in magnetostatics. ESAIM: M2AN 51 (2017) 615–640. | DOI | Numdam | MR | Zbl

K. Yosida, Functional Analysis. Springer, Heidelberg (1980). | Zbl

I. Yousept, Finite Element Analysis of an Optimal Control Problem in the Coefficients of Time-Harmonic Eddy Current Equations. J. Optimiz. Theory Appl. 154 (2012) 879–903. | DOI | MR | Zbl

I. Yousept, Optimal control of Maxwell’s equations with regularized state constraints. Comput. Optimiz. Appl. 52 (2012) 559–58. | DOI | MR | Zbl

I. Yousept, Optimal Control of Quasilinear H(curl)-Elliptic Partial Differential Equations in Magnetostatic Field Problems. SIAM J. Control Optim. 51 (2013) 3624–3651. | DOI | MR | Zbl

Cité par Sources :