An iterative FEM with fast multipole updates for scattering from an electically large object

As known widely, the finite element method(FEM) has been widely used as a powerful tool for solving numerous electromagnetic problems since it is able to deal with arbitrary geometries and inhomogeneous and complex materials. Besides, the FEM gets used for solving scattering problems as well as typical bounded problems through hybridization with other methods such as a boundary integral method(BIM) or incorporation with an appropriate absorbing boundary condition. Among these methods for applying the FEM to scattering problems, the iterative FEM has been proposed(1-2) and complemented(3) by eliminating the internal resonance(4) in the recent years. According to this method, the FEM gives an efficient and accurate result with only a small number of meshes near around a scatterer through several iterative updates of the boundary conditions. The generated system matrix does not only preserve sparsity, which is a very good property of the typical FEM, but is also invariant during the iterations. Besides, the matrix for updating the boundary conditions is invariant as well. Thus, the numerical calculation for these two matrices is performed only once at the first iteration. These properties make this method efficient and competitive with other methods such as a finite element - boundary integral method(FEBIM). However, this method still has a bottleneck that the updating matrix is a full matrix. Just as the method of moments, this property makes it difficult to apply the iterative FEM to scattering by an electrically large object. Assuming that the numbers of unknowns on the fictitious boundary and on the boundary where the equivalent current is calculated are NF and NM, respectively, both the operation counts for matrix-vector multiplication per iteration and the memory requirements for storage of the updating matrix will be proportional to O(NFNM). If NF is almost equal to NM, they will be O(NF 2 ). Therefore, in this paper, in order to apply the iterative FEM to a large-body problem, we develop a scheme to reduce both the operation counts and the memory requirements to a lower order by applying the well known fast multipole method(FMM) to the boundary updating procedure.