Iterative solution of boundary element approximations for electric and magnetic fields

The calculation of potentials in homogeneous and isotropic media by the boundary element method has the advantage that a harmonic solution of Laplace's equation is obtained for given approximated boundary conditions. The technique leads to the solution of linear systems with full matrices of dimension 1000–10,000 for medium- and large-sized three-dimensional problems, An efficient solution procedure of the linear systems is required. While the iterative solution of the large and sparse linear systems arising from the finite difference or the finite element method is well documented, the systems resulting from the boundary element method are typically solved by direct methods. However, in many cases an iterative solver needs far fewer operations to achieve a sufficient accuracy. Importantly, there are many alternative methods, each of them well suited for different types of problem. Here, we provide an overview of state-of-the-art iterative solvers. We will discuss the particular methods that have been successfully applied to systems arising from field calculations in the high-voltage engineering by the boundary element method. The selection of appropriate methods is discussed. We demonstrate that iterative solutions can be much faster than direct solvers with regards to the number of operations. Furthermore, these solvers are optimally suited for today's supercomputers because they can be efficiently vectorized and parallelized.