New highly efficient singularity cancellation methods are proposed for the surface inegral equations (SIEs), where general formulations are presented for a series of radial-angular transformations. The transformation equations are effective for higher order of SIE basis functions, which are utilized for both near couplings and the multilevel fast multipole method (MLFMM). Higher order of basis functions are utilized for the hybrid finite element boundary integral (FE-BI) equations according to the electromagnetic boundary conditions, where the FE-BI method combines the finite element method (FEM) and the boundary integral (BI) equations. Exellent accuracy of the FE-BI method is obtained using a variety of EM simulation examples. Analytical solutions to the system matrix of the FEM are provided based on the simplex coordinates as a research reference.