Preconditioning methods for eddy-current optimally controlled time-harmonic electromagnetic problems
At first, the Maxwell's equations for finding a time-dependent magnetic vector potential in a bounded three-dimensional domain are given. It is described that the finite element discretization of the considered problem leads to a system of linear equations, where the matrix \(\mathcal{A}\) has a two-by-two block structure. Then, a preconditioner \(\mathcal{C}\) for such systems of equations is presented and estimates of the eigenvalues of the matrix \(\mathcal{C}^{-1}\mathcal{A}\) are derived. As an application of the proposed preconditioner, an optimal control problem of parabolic type with a time-harmonic target state is discussed. Furthermore, an eddy current electromagnetic problem is considered. Its discretization leads to a matrix with a double two-by-two block structure. The construction of a preconditioner is described for two cases, namely, the situation where the control and observation functions are defined on the whole space and the situation where these functions are prescribed only on a subset of the whole domain. Numerous numerical tests are presented, which show the robustness and performance of the iterative solver with the new preconditioners.
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