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.
Accepté le :
DOI : 10.1051/m2an/2016030
Mots clés : Optimal control, magnetostatic Maxwell equation, a posteriori error estimate, edge element, adaptive convergence
@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://www.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://www.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://www.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://www.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
Residual based a posteriori error estimators for eddy current computation. ESAIM: M2AN 34 (2000) 159–182. | DOI | Numdam | MR | Zbl
, , and ,Axioms of adaptivity. Comp. Math. Appl. 67 (2014) 1195–1253. | DOI | MR | Zbl
, , and ,Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46 (2008) 2524–2550. | DOI | MR | Zbl
, , and ,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
, and ,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
, and ,An adaptive multilevel method for time-harmonic Maxwell equations with singularities. SIAM J. Sci. Comput. 29 (2007) 118–138. | DOI | MR | Zbl
, and ,P.G. Ciarlet, Finite Element Methods for Elliptic Problems. North-Holland, Amsterdam (1978). | MR | Zbl
Edge element methods for Maxwell’s equations with strong convergence for Gauss’ laws. SIAM J. Numer. Anal. 52 (2014) 779–807. | DOI | MR | Zbl
, and ,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
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
, , and ,Convergence of adaptive edge element methods for the 3D currents equations. J. Comput. Math. 27 (2009) 657–676. | DOI | MR | Zbl
and ,Adaptive edge element approximation of -elliptic optimal control problems with control constraints. BIT Numer. Math. 55 (2015) 255–277. | DOI | MR
and ,A recursive approach to local mesh refinement in two and three dimensions. J. Comp. Appl. Math. 55 (1995) 275–288. | DOI | MR | Zbl
,A posteriori error estimates for distributed convex optimal control problems. Adv. Comput. Math. 15 (2001) 285–309. | DOI | MR | Zbl
and ,P. Monk, Finite Element Methods for Maxwell’s Equations. Oxford University Press, New York (2003). | MR | Zbl
A basic convergence result for conforming adaptive finite elements. Math. Models Methods Appl. Sci. 18 (2008) 707–737. | DOI | MR | Zbl
, and ,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
A posteriori error estimates for Maxwell equations. Math. Comput. 77 (2008) 633–649. | DOI | MR | Zbl
,Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54 (1990) 483–493. | DOI | MR | Zbl
and ,A convergence proof for adaptive finite elements without lower bounds. IMA J. Numer. Anal. 31 (2011) 947–970. | DOI | MR | Zbl
,An algorithm for adaptive mesh refinement in dimensions. Computing 59 (1997) 115–137. | DOI | MR | Zbl
,PDE-constrained optimization of time-dependent 3D electromagnetic induction heating by alternating voltages. ESAIM: M2AN 46 (2012) 709–729. | DOI | Numdam | MR | Zbl
and ,R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, New York (1996). | Zbl
A local compactness theorem for Maxwell’s equations. Math. Methods Appl. Sci. 2 (1980) 12–25. | DOI | MR | Zbl
,Convergence of an adaptive finite element method for distributed flux reconstruction. Math. Comput. 84 (2015) 2645–2663. | DOI | MR | Zbl
and ,Optimal control of Maxwell’s equations with regularized state constraints. Comput. Optim. Appl. 52 (2012) 559–581. | DOI | MR | Zbl
,Optimal control of quasilinear -elliptic partial differential equations in magnetostatic field problems. SIAM J. Control Optimiz. 51 (2013) 3624–3651. | DOI | MR | Zbl
,Convergence and optimality of adaptive edge finite element methods for time-harmonic Maxwell equations. Math. Comput. 81(2012) 623–642. | DOI | MR | Zbl
, , , and ,Cité par Sources :