Helmholtz subspaces preserving fast solvers based on multigrid inversions of loop-star decompositions

The discretized electric field integral equation operator becomes severely ill-conditioned when the frequency is low or the discretization is dense. Calderón preconditioning or hierarchical bases can be used to prevent these breakdowns. Difficulties arise, however, when these schemes are combined with fast multiplication methods. In fact the scalar potential part of the electric field integral operator on the solenoidal Helmholtz subspace of the electric current is set to zero only approximatively by the fast method. This contribution presents an O(n log n) fast solver that preserves the Helmholtz subspace independently of the compression level and accuracy. The new scheme is based on the inversion of the loop-star decomposition and it is purely algebraic in nature.