Fast direct solution of FEM systems using overlapped localizing modes on a shifted grid

Overlapped, localizing local-global solution (OL-LOGOS) modes have been proposed to develop fast direct solvers for low frequency electromagnetic wave problems. The efficiencies of the resulting OL-LOGOS factorization algorithms have been demonstrated for the matrix equations associated with dense three-dimensional integral equations and sparse two-dimensional partial differential equations. In both cases, approximately O(N log N) time and O(N) memory complexities have been observed. In this work, the OL-LOGOS method is applied to three-dimensional scalar FEM systems. In order to improve the factorization speed and reduce memory costs for FEM applications, a pre-factorization permutation step is incorporated into the OL-LOGOS factorization algorithm. Numerical results demonstrate factorization and memory complexities of approximately O(N log N) and O(N) as the problem size grows.