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Summary form only given, as follows. An implementation of the Jacobi-Davidson method for solving non-Hermitian eigenproblems in electromagnetics is presented. The standard Jacobi-Davidson algorithm has been extended to reduce memory requirements and to improve computational speed through the use of a multigrid approach. Algorithmic details particular to the types of problem typically encountered in design applications are described, including the capability to handle highly lossy materials and the need for many degrees of freedom to accurately represent an electromagnetic field. The classes of eigenproblem to which this method has been applied include periodic and lossy structures, discretized on both structured and unstructured meshes. Each class of problem are described, with reference to the particular difficulties that it presents, and examples are used to demonstrate the effectiveness of this technique as a general eigenfrequency solver.