Geodesic
A Convergent adaptive edge element method for an optimal control problem in magnetostatics
Xu, Yifeng1 ; Zou, Jun2
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.
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 2, pp. 615-640

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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.

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DOI : 10.1051/m2an/2016030
Classification : 65N12, 65N15, 65N30, 35Q60, 49K20, 49M05
Keywords: 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. 
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     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://geodesic.mathdoc.fr/articles/10.1051/m2an/2016030/}
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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

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