Studies on preconditioning and basis functions in periodic fast multipole methods for Maxwell's equations

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This study aims at further accelerating the periodic FMM, which is a fast method for solving electromagnetic scattering problems for periodic structures, by improving preconditioners for linear equations, basis functions and integral equation formulations. The square of certain integral operators in electromagnetic scattering problems are well-conditioned, and so are their numerical counterparts as one uses the right basis functions. We were able to obtain an efficient solver of periodic scattering problems using this idea. We also investigated other well-conditioned integral equation formulations and developed a solution method for almost periodic structures found in photonic crystal applications, as well as an efficient preconditioner for volume integral equations.