A Convergent adaptive edge element method for an optimal control problem in magnetostatics
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 2, pp. 615-640.

This work is concerned with an adaptive edge element solution of an optimal control problem associated with a magnetostatic saddle-point Maxwell’s system. An a posteriori error estimator of the residue type is derived for the lowest-order edge element approximation of the problem and proved to be both reliable and efficient. With the estimator and a general marking strategy, we propose an adaptive edge element method, which is demonstrated to generate a sequence of discrete solutions converging strongly to the exact solution satisfying the resulting optimality conditions and guarantee a vanishing limit of the error estimator.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2016030
Classification : 65N12, 65N15, 65N30, 35Q60, 49K20, 49M05
Mots clés : Optimal control, magnetostatic Maxwell equation, a posteriori error estimate, edge element, adaptive convergence
Xu, Yifeng 1 ; Zou, Jun 2

1 Department of Mathematics, Scientific Computing Key Laboratory of Shanghai Universities and E-Institute for Computational Science of Shanghai Universities, Shanghai Normal University, Shanghai 200234, P.R. China.
2 Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong, P.R. China.
@article{M2AN_2017__51_2_615_0,
     author = {Xu, Yifeng and Zou, Jun},
     title = {A {Convergent} adaptive edge element method for an optimal control problem in magnetostatics},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {615--640},
     publisher = {EDP-Sciences},
     volume = {51},
     number = {2},
     year = {2017},
     doi = {10.1051/m2an/2016030},
     mrnumber = {3626413},
     zbl = {1366.78022},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2016030/}
}
TY  - JOUR
AU  - Xu, Yifeng
AU  - Zou, Jun
TI  - A Convergent adaptive edge element method for an optimal control problem in magnetostatics
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2017
SP  - 615
EP  - 640
VL  - 51
IS  - 2
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an/2016030/
DO  - 10.1051/m2an/2016030
LA  - en
ID  - M2AN_2017__51_2_615_0
ER  - 
%0 Journal Article
%A Xu, Yifeng
%A Zou, Jun
%T A Convergent adaptive edge element method for an optimal control problem in magnetostatics
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2017
%P 615-640
%V 51
%N 2
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an/2016030/
%R 10.1051/m2an/2016030
%G en
%F M2AN_2017__51_2_615_0
Xu, Yifeng; Zou, Jun. A Convergent adaptive edge element method for an optimal control problem in magnetostatics. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 2, pp. 615-640. doi : 10.1051/m2an/2016030. http://archive.numdam.org/articles/10.1051/m2an/2016030/

M. Ainsworth and J.T. Oden, A Posteriori Error Estimation in Finite Element Analysis. Pure and Applied Mathematics. Wiley-Interscience, New York (2000). | MR | Zbl

R. Beck, R. Hiptmair, R.H.W. Hoppe and B. Wohlmuth, Residual based a posteriori error estimators for eddy current computation. ESAIM: M2AN 34 (2000) 159–182. | DOI | Numdam | MR | Zbl

C. Carstensen, M. Feischl, M. Page and D. Praetorius, Axioms of adaptivity. Comp. Math. Appl. 67 (2014) 1195–1253. | DOI | MR | Zbl

J.M. Cascon, C. Kreuzer, R.H. Nochetto and K.G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46 (2008) 2524–2550. | DOI | MR | Zbl

J. Chen, Y. Xu and J. Zou, An adaptive edge element method and its convergence for a saddle-point problem from magnetostatics. Numer. Methods PDEs 28 (2012) 1643–1666. | DOI | MR | Zbl

Z. Chen, Q. Du and J. Zou, Finite element methods with matching and nonmatching meshes for Maxwell equations with discontinuous coefficients. SIAM J. Numer. Anal. 37 (2000) 1542–1570. | DOI | MR | Zbl

Z. Chen, L. Wang and W. Zheng, An adaptive multilevel method for time-harmonic Maxwell equations with singularities. SIAM J. Sci. Comput. 29 (2007) 118–138. | DOI | MR | Zbl

P.G. Ciarlet, Finite Element Methods for Elliptic Problems. North-Holland, Amsterdam (1978). | MR | Zbl

P. Ciarlet, Jr., H. Wu and J. Zou, Edge element methods for Maxwell’s equations with strong convergence for Gauss’ laws. SIAM J. Numer. Anal. 52 (2014) 779–807. | DOI | MR | Zbl

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

M. Hintermüller, R.H.W. Hoppe, Y. Iliash and M. Kieweg, An a posteriori error analysis of adaptive finite element methods for distributed elliptic control problems with control constraints. ESAIM: COCV 14 (2008) 540–560. | Numdam | MR | Zbl

R.H.W. Hoppe and J. Schöberl, Convergence of adaptive edge element methods for the 3D currents equations. J. Comput. Math. 27 (2009) 657–676. | DOI | MR | Zbl

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

I. Kossaczky, A recursive approach to local mesh refinement in two and three dimensions. J. Comp. Appl. Math. 55 (1995) 275–288. | DOI | MR | Zbl

W. Liu and N. Yan, A posteriori error estimates for distributed convex optimal control problems. Adv. Comput. Math. 15 (2001) 285–309. | DOI | MR | Zbl

P. Monk, Finite Element Methods for Maxwell’s Equations. Oxford University Press, New York (2003). | MR | Zbl

P. Morin, K.G. Siebert and A. Veeser, A basic convergence result for conforming adaptive finite elements. Math. Models Methods Appl. Sci. 18 (2008) 707–737. | DOI | MR | Zbl

R.H. Nochetto, K.G. Siebert and A. Veeser, Theory of adaptive finite element methods: an introduction. Multiscale, Nonlinear and Adaptive Approximation, edited by R.A. DeVore and A. Kunoth. Springer, New York (2009) 409–542. | MR | Zbl

J. Schöberl, A posteriori error estimates for Maxwell equations. Math. Comput. 77 (2008) 633–649. | DOI | MR | Zbl

L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54 (1990) 483–493. | DOI | MR | Zbl

K.G. Siebert, A convergence proof for adaptive finite elements without lower bounds. IMA J. Numer. Anal. 31 (2011) 947–970. | DOI | MR | Zbl

C. Traxler, An algorithm for adaptive mesh refinement in n dimensions. Computing 59 (1997) 115–137. | 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

R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, New York (1996). | Zbl

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

Y. Xu and J. Zou, Convergence of an adaptive finite element method for distributed flux reconstruction. Math. Comput. 84 (2015) 2645–2663. | DOI | MR | Zbl

I. Yousept, Optimal control of Maxwell’s equations with regularized state constraints. Comput. Optim. Appl. 52 (2012) 559–581. | DOI | MR | Zbl

I. Yousept, Optimal control of quasilinear 𝐇(𝐜𝐮𝐫𝐥)-elliptic partial differential equations in magnetostatic field problems. SIAM J. Control Optimiz. 51 (2013) 3624–3651. | DOI | MR | Zbl

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

Cité par Sources :