Improving the Efficiency of Three-Dimensional Multilevel Fast Multipole Algorithm by the Triangular Interpolation

The efficiency of the multilevel fast multipole algo-rithm (MLFMA), which needs the interpolation and anterpola-tion (I&A) processes between consecutive levels, is improved by using the isosceles triangular interpolation (TI) instead of the commonly used Lagrange interpolation (LI). For the pth-order interpolation where p > 0, the number of sampling data points used for the TI is ( p + 1)( p + 2)/2, much less than the LI, which needs (p + 1)(2) points. To effectively use the TI, the sampling points on the Ewald sphere should be reselected to form isosceles triangular data grids but not the common rectangular ones. However, the numerical integration in the ?-direction over the Ewald sphere is based on the Gauss-Legendre quadrature rule, resulting in nonuniform triangular grids. A novel TI approach suitable for the nonuniform grids is proposed, while the interpolation error in the TI is analyzed in detail. Numerical results of typical electromagnetic (EM) scattering cases are shown to illustrate the accuracy and efficiency of the proposed TI-based MLFMA.