Edge Element Method for Optimal Control of Stationary Maxwell System with Gauss Law

A novel edge element method is proposed for the optimal control of the stationary Maxwell system with a nonvanishing charge density. The proposed approach does not involve the usual saddle-point formulation and features a positive definite structure in the associated equality constraints, for which optimal preconditioners are available in combination with conjugate gradient iteration. Our main results include error estimates and strong convergence for both the optimal edge element solution and the associated discrete Gauss laws. In particular, our analysis helps improve significantly the convergence rate established by Ciarlet, Wu, and Zou [SIAM J. Numer. Anal., 52 (2014), pp. 779--807] for the edge element method for the stationary Maxwell system. Numerical experiments are presented to verify the theoretical results.

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