Low-frequency multilevel fast multipole algorithm using an approximate diagonalization of the Green´s function

We present an approximate diagonalization of the Green´s function to implement a stable multilevel fast multipole algorithm (MLFMA) for low-frequency problems. The diagonalization is based on scaled spherical functions, leading to stable computations of translation operators at all distances and for all frequencies. Similar to the conventional diagonalization, shift operators are expressed in terms of complex exponentials, while radiated and incoming fields are expanded in terms of scaled plane waves. Even though its accuracy is limited, the low-frequency MLFMA developed by using the proposed diagonalization technique provides stable matrix-vector multiplications for arbitrarily low frequencies, while it can easily be implemented via minor modifications on the existing codes.