A combination of the fast multipole boundary element method and Krylov subspace recycling solvers

The solution of the Helmholtz equation by the Boundary Element Method leads to a sequence of frequency dependent linear systems of equations, where each is typically solved independently. The Krylov Subspace Recycling algorithms, like the GCRO-DR and the GCROT, are based on the idea that the solutions of consecutive systems have similarities and the information of the previous cycle can be reused to accelerate the convergence. These solvers showed very good results for sparse matrices arising in the FEM and are now applied to the fully populated BEM matrices. Additionally, the solution of a single system of equations is accelerated by the Fast Multipole Method, which shows a mostly linear correlation between iterations and calculation time. Hence the newly proposed combination has a high potential of achieving a faster solution process. The 3D Fast Multipole Boundary Element Method additionally incorporates a Burton–Miller formulation and a halfspace formulation to be applicable to a wider range of engineering problems. The method is illustrated and discussed by two different numerical examples. The advantages and critical aspects of the combination are presented.