Parallel hierarchical decomposition of finite element method with block diagonal symmetric Gauss-Seidel preconditioner for solving 3D problems with over ten billion unknowns

We present in this paper an efficient parallelization approach of the finite element domain decomposition method (FEM-DDM) for solving large-scale 3D scattering problems with over ten billion unknowns. In this parallel FEM-DDM, the whole solution domain is decomposed in a two-level hierarchical way to minimize the inter process communication. A communication-avoided block diagonal symmetric Gauss-Seidel (BD-SGS) preconditioner is specially designed to accelerate the iterative solution of the interface problem from FEM-DDM and reduce the CPU time, as well as the peak memory. Besides, a communication strategy is carefully designed to make computation and communication overlapped via nonblocking sending and receiving to further improve the parallel scalability. Numerical examples are presented to demonstrate the accuracy, scalability, and capability of the proposed parallel FEM-DDM algorithm. The solutions of scattering by a practical aircraft model with a three-layer antenna radome on its nose section is presented, which are discretized with more than ten billion unknowns. To the best of our knowledge, this is the largest number of unknowns solved by parallel FEM-DDM in computational electromagnetics to date, for this type of simulation.