What is it about?
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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Perspectives
Instead of the discrete compactness of Nedelec edge elements, our approach to convergence of finite element approximations over locally refined meshes depends on the weak convergence of discrete states and controls as well as a simple observation: the sequence of minima to discrete objective functionals converges to the minimum of a functional with respect to some auxiliary optimization problem resulting from the adaptive algorithm.
Dr Yifeng Xu
Shanghai Normal University
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This page is a summary of: A Convergent adaptive edge element method for an optimal control problem in magnetostatics, ESAIM Mathematical Modelling and Numerical Analysis, February 2017, EDP Sciences,
DOI: 10.1051/m2an/2016030.
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