An H-LU Preconditioner for the Hybrid Finite Element-Bomdaty Integral

A flexible and efficient $\mathcal{H}$-LU-based preconditioner ($\mathcal{H}$-LU-P) is presented for the hybrid finite element-boundary integral-multilevel fast multipole algorithm (FE-BI-MLFMA) for solving 3D scattering by inhomogeneous objects in this paper. The formulation of FE-BI is firstly approximated by using locally approximated integral operators for the BI part to construct a FEM-ABC based precondition matrix. Then the precondition matrix equation is solved by the nested dissection (ND) accelerated $\mathcal{H}$-LU-based fast direct solver. Performance of the $\mathcal{H}$-LU-P is studied numerically for different problems, including the quasi-static problem, 2D extended and 3D extended electrodynamic problems, etc. Numerical experiments show the $\mathcal{H}$-LU-P has an O(NlogN) memory complexity and an O(Nlog2N) CPU time complexity for the quasi-static, the 2D extended lossless and the 3D extended lossy problems. For the 3D extended lossless problems, the complexity is larger due to the increasing rank of the $\mathcal{H}$-LU, but it still outperforms alternative direct solvers, such as the popular multifrontal-based solver MUMPS. Large realistic scattering problems with more than ten million unknowns are calculated, including a honeycomb structure with 8100 elements, showing the capability and efficiency of our proposed preconditioner.