Boundary control of time-harmonic eddy current equations

Motivated by various applications, this article develops the notion of boundary control for Maxwell's equations in the frequency domain. Surface curl is shown to be the appropriate regularization in order for the optimal control problem to be well-posed. Since, all underlying variables are assumed to be complex valued, the standard results on differentiability do not directly apply. Instead, we extend the notion of Wirtinger derivatives to complexified Hilbert spaces. Optimality conditions are rigorously derived and higher order boundary regularity of the adjoint variable is established. The state and adjoint variables are discretized using higher order N\'ed\'elec finite elements. The finite element space for controls is identified, as a space, which preserves the structure of the control regularization. Convergence of the fully discrete scheme is established. The theory is validated by numerical experiments, in some cases, motivated by realistic applications.Comment: 25 pages, 6 figure

Local decay rates of best-approximation errors using vector-valued finite elements for fields with low regularity and integrable curl or divergence

International audienceWe estimate best-approximation errors using vector-valued finite elements for fields with low regularity in the scale of the fractional-order Sobolev spaces. By assuming that the target field enjoys an additional integrability property on its curl or its divergence, we establish upper bounds on these errors that can be localized to the mesh cells. These bounds are derived using the quasi- interpolation errors with or without boundary prescription derived in [A. Ern and J.-L. Guermond, ESAIM Math. Model. Numer. Anal., 51 (2017), pp. 1367–1385]. In the present work, a localized upper bound on the quasi-interpolation error is derived by using the face-to-cell lifting operators analyzed in [A. Ern and J.-L. Guermond, Found. Comput. Math., (2021)] and by exploiting the additional assumption made on the curl or the divergence of the target field. As an illustration, we show how to apply these results to the error analysis of the curl-curl problem associated with Maxwell’s equations

Integrated design of chemical waste water treatment systems.

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