| Issue |
ESAIM: M2AN
Volume 55, Number 5, September-October 2021
|
|
|---|---|---|
| Page(s) | 2013 - 2044 | |
| DOI | https://doi.org/10.1051/m2an/2021046 | |
| Published online | 29 September 2021 | |
An adaptive edge element method and its convergence for an electromagnetic constrained optimal control problem
1
Department of Mathematics, Duke University, Durham, NC 27708, USA
2
Department of Mathematics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong
* Corresponding author: zou@math.cuhk.edu.hk
Received:
8
August
2020
Accepted:
17
August
2021
In this work, an adaptive edge element method is developed for an H(curl)-elliptic constrained optimal control problem. We use the lowest-order Nédélec’s edge elements of first family and the piecewise (element-wise) constant functions to approximate the state and the control, respectively, and propose a new adaptive algorithm with error estimators involving both residual-type error estimators and lower-order data oscillations. By using a local regular decomposition for H(curl)-functions and the standard bubble function techniques, we derive the a posteriori error estimates for the proposed error estimators. Then we exploit the convergence properties of the orthogonal L2-projections and the mesh-size functions to demonstrate that the sequences of the discrete states and controls generated by the adaptive algorithm converge strongly to the exact solutions of the state and control in the energy-norm and L2-norm, respectively, by first achieving the strong convergence towards the solution to a limiting control problem. Three-dimensional numerical experiments are also presented to confirm our theoretical results and the quasi-optimality of the adaptive edge element method.
Mathematics Subject Classification: 65K10 / 65N12 / 65N15 / 65N30 / 49J20
Key words: Constrained optimal control / Maxwell’s equations / a posteriori error estimates / adaptive edge element method / convergence analysis of adaptive algorithm
© The authors. Published by EDP Sciences, SMAI 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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