A direct hierarchical multilevel preconditioner for the solution of finite element-boundary integral equations

Boundary integral (BI) equations in combination with fast solvers such as the multilevel fast multipole method are very well suited for solving electromagnetic scattering and radiation problems. In order to consider dielectric objects, the BI approach can be extended to the hybrid finite element-boundary integral (FE-BI) method. By using hierarchical higher order basis functions for the expansion of the unknowns, very accurate results can be obtained. To accelerate the convergence of iterative solvers, the usage of preconditioning methods is indispensable. In this work, a very efficient multilevel preconditioner is presented which is based on a factorization of the submatrices containing the interactions of the zeroth order divergence-conforming basis functions in the BI formulation and the corresponding lowest order curl-conforming basis of the FE method. Moreover, it is shown that the number of fill-ins can be effectively reduced by exploiting advanced reordering algorithms.