Abstract
In this work, an adaptive edge element method is developed for anH(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 forH(curl)-functions and the standard bubble function techniques, we derive thea posteriorierror estimates for the proposed error estimators. Then we exploit the convergence properties of the orthogonalL2-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 andL2-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.
Highlights
Many electromagnetic simulation problems involve the following H(curl)-elliptic equation [13, 44]: curl(︀μ−1curl y)︀ + σy = f in Ω, (1.1)where σ and μ are the electric permittivity and the magnetic permeability, respectively
We are interested in a relevant optimal control problem, namely to find a specially designed external current source profile so that the resulting electromagnetic field achieves an optimal target design
We show that the maximal error estimators and the residuals corresponding to the sequence of adaptive states and adjoint states vanish
Summary
Constrained optimal control, Maxwell’s equations, a posteriori error estimates; adaptive edge element method, convergence analysis of adaptive algorithm. 4.6 and 4.7), where we use a generalized convergence result of the mesh-size functions (cf (4.4)), instead of introducing the buffer layer (cf [61]) between the meshes at different levels, so that our proof can be significantly simplified By means of these auxiliary results and again the properties of L2-projections, we are able to prove that the limit point of the discrete triplets {(y*k, p*k, u*k)} solves the variational inequality associated with the control problem (cf Thm. 4.8). If x ≤ Cy and x ≥ Cy hold simultaneously, we denote it by x ≈ y
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