Iterative Approaches to the Solution of Electromagnetic Boundary Value Problems

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Abstract
The objective of this paper is to present the development of the conjugate gradient method (CGM). and other related iterative techniques, by viewing the iterative problem as that of reducing the norm of the error in the satisfaction of the boundary conditions in a systematic manner. It is demonstrated that the choice of the direction vectors, as dictated by CGM. is not optimal and that alternate choices for these vectors, that have the potential of accelerating the convergence over that achieved by the CGM technique, are theoretically possible. It is also shown that the approximate inverse of the operator, which can be constructed by using the spectral iterative technique, can some-times be employed with advantage, to generate these direction vectors. The important problem of multiple incident fields is addressed and the difficulties associated with the CGM method for multiple right-hand side problem is linked lo the machine round-off errors that are responsible for the loss of orthogonality generated in a recursive manner according to CGM. Illustrative numerical examples based on the use of CGM, approximate inverse and other related approaches are included in the paper.