The Multilevel Fast Multipole Algorithm (MLFMA) for Solving Large-Scale Computational Electromagnetics Problems

3.6.3 Iterative Solutions of Dielectric Objects

For electromagnetics problems involving dielectric objects, two types of efficient preconditioners, namely, the two-partition block-diagonal preconditioner (2PBDP) and the four-partition block-diagonal preconditioner (4PBDP), can be employed to reduce the number of iterations required for solutions. Matrix equations in the form of (2.40) can be preconditioned as

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where is a preconditioner matrix. In MLFMA, there are near-field interactions, which are calculated directly and are available for constructing preconditioners. BDP, which is based on using self-interactions of boxes at the lowest level, is commonly used to accelerate MLFMA solutions of electromagnetics problems involving PEC objects formulated with MFIE and CFIE [35]. The preconditioner matrix, which has a block-diagonal structure, can be inverted and used efficiently with complexity. A direct extension of BDP for dielectric problems, which is called 2PBDP, involves self-interactions of boxes at the lowest level in diagonal partitions, i.e.,

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where and are block-diagonal ...