Helmholtz-stable fast solution of the Combined Field Integral Equation

This work presents a new fast matrix-vector multiplication scheme for the solution of the Combined Field Integral Equation. Similarly to other fast methods, our approach reduces the matrix-vector multiplication cost from O(N²) to O(N log N). Differently from other fast solvers, however, the effectiveness of CFIE preconditioning techniques such as quasi-Helmholtz decompositions or Calderón approaches is maintained by our method even for very high matrix compression rates. This is thanks to the fact that, in the scheme we are proposing, the contribution from the scalar potential of the electric field operator, when applied to or tested with solenoidal functions is always zero independent of the compression error. In addition, the new method will take advantage of the redundancies of the electric and magnetic integral kernels to store the magnetic part of the CFIE at a minimal additional cost with respect to the cost of the electric part. Numerical results will show the effectiveness of our approach and its impact on the solution of realistic problems.