An Effective $\mathcal{H}$-LU-Based Preconditioner for the FE-BI-MLFMA for 3-D Scattering Problems

Abstract

A flexible and efficient Hierarchical LU ( $\mathcal{H}$ -LU)-based preconditioner ( $\mathcal{H}$ -LU-P) is presented for the hybrid finite element-boundary integral (FE-BI) multilevel fast multipole algorithm for solving three-dimensional scattering by inhomogeneous objects. The formulation of FE-BI is first approximated by using locally approximated integral operators for the BI part to construct a finite element method-absorbing boundary condition (FEM-ABC) based preconditioning matrix. Then, the preconditioning matrix equation is solved by the nested dissection accelerated $\mathcal{H}$ -LU-based fast direct solver. A semi-geometric clustering strategy is proposed to simplify the cluster tree-building procedure in representing the preconditioning matrix in the $\mathcal{H}$ -matrix form. Numerical results demonstrate that the $\mathcal{H}$ -LU-P outperforms other alternative preconditioners, based on multifrontal approach. The scattering by a large and complex aircraft model with a radome on its nose section is calculated, showing the capability and efficiency of the preconditioner.