For a Helmholtz eigenvalue problem with a multiply connected domain, the boundary integral equation approach as well as the boundary-element method is shown to yield spurious eigenvalues even if the complex-valued kernel is used. In such a case, it is found that spurious eigenvalues depend on the geometry of the inner boundary. Demonstrated as an analytical case, the spurious eigenvalue for a multiply connected problem with its inner boundary as a circle is studied analytically. By using the degenerate kernels and circulants, an annular case can be studied analytically in a discrete system and can be treated as a special case. The proof for the general boundary instead of the circular boundary is also derived. The Burton{Miller method is employed to eliminate spurious eigenvalues in the multiply connected case. Moreover, a modi ed method considering only the real-part formulation is provided. Five examples are shown to demonstrate that the spurious eigenvalues depend on the shape of the inner boundary. Good agreement between analytical prediction and numerical results are found.

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For MEMS combdrive design, the reduction of levitating force due to electrostatic fields is very important, and an accurate electrostatic analysis is essential and indispensable. For diverse MEMS combdrive designs, the boundary element method (BEM) has become a better method than the domain-type finite element method (FEM) because the BEM can provide a complete solution in terms of boundary values only, with substantial saving in modeling effort. Since dual BEM (DBEM) has some advantages over conventional BEM for a singularity, the DBEM was used to simulate the fringing of field around the edges of the fixed fingers and movable fingers of MEMS combdrives for diverse design cases. A number of electrostatic problems for typical MEMS combdrive designs were analyzed to check the efficiency and validity of this new technique. It is found that the numerical results computed by coarse mesh DBEM match the reference data from a large refined mesh FEM very well, and the accuracy and performance of DBEM are also better than those of conventional BEM for solving the electric intensity field of MEMS combdrives. By way of the DBEM presented in this paper, an accurate and reasonable electrostatic field can be obtained, and the follow-up control method of levitating force for the MEMS combdrive can be implemented more precisely.