Combining Calderón preconditioning with Fast Multipole Methods

Abstract

The Method of Moments (MoM) is one of the most popular techniques for solving electromagnetic scattering problems. Its main advantage is that, when used to discretise a Boundary Integral Equation (BIE), only the surface of the objects needs to be discretised. Many other techniques require discretisation of the volume as well, leading to an increase of unknowns and requiring an artificial boundary to emulate infinity. The downside of the MoM is that the resulting linear system is dense, due to the Green's function, and often ill-conditioned. The density of the impedance matrix would lead to a prohibitive O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) complexity for both the CPU time and the memory. A number of techniques have been developed in the past, reducing this complexity. Among the most succesful are the Fast Multipole Methods (FMM), which reduce the complexity for a matrix-vector product to O (N log N). This matrix-vector product is the bottleneck of an iterative solution of the linear system. However, an iterative solution can only be efficient if the problem is sufficiently well-conditioned. The choice of BIE influences the conditioning number of the impedance matrix. The Electric Field Integral Equation (EFIE) leads to a highly accurate solution but is notoriously ill-conditioned. In comparison, the Magnetic Field Integral Equation (MFIE) is less accurate (for the same level of discretisation) but is well-posed. Also, on open surfaces the MFIE is not applicable. In order to alleviate the problems with the EFIE, a powerful preconditioner is required to stabilise the breakdown, which occurs in particular at low frequencies (LF). Recently, an approach has been proposed, allowing multiplication of the EFIE impedance matrix with itself, leading to a well-conditioned problem. In the next section, this approach will be briefly revisited. In order to maintain the O (N log N) complexity for the global solution, we employ FMM to accelerate the Calderon Preconditioner as well. A number of difficulties occur, but these can all be overcome. This combination of FMM and Calderon Preconditioning will be dealt with in the remaining sections. At the time of the conference, examples will be shown to illustrate the novel techniques.