Preconditioners for time-harmonic optimal control eddy-current problems

Abstract

Time-harmonic formulations enable solution of time-dependent PDEs without use of normally slow time-stepping methods. Two efficient preconditioners for the discretized parabolic and eddy current electromagnetic optimal control problems, one on block diagonal form and one utilizing the two by two block structure of the resulting matrix, are presented with simplified analysis and numerical illustrations. Both methods result in tight eigenvalue bounds for the preconditioned matrix and very few iterations that hold uniformly with respect to the mesh, problem and method parameters, with the exception of the dependence on reluctivity for the block diagonal preconditioner.