A Robust and Scalable Multigrid Solver for 3-D Low-Frequency Electromagnetic Diffusion Problems

Multigrid (MG) solvers, typically increasing the computational time linearly with the grid size, are suitable for large-scale forward modeling problems. However, for electromagnetic (EM) problems as frequency decreases and the grid is increasingly stretched, MG solvers for EM modeling can converge slowly or even diverge. We propose an efficient four-color line Gauss-Seidel (GS) MG for finite difference (FD) frequency-domain EM solution. In this algorithm, the edge components attached to nodes on each line in one particular direction are updated simultaneously, leading to the solution in each local region satisfies divergence-free condition. Due to the fact that each local linear system of equations is completely uncoupled with that formed for its disjoint lines, we can group all lines of grid nodes into four colors with the requirement that all local systems formed for lines with the same color are disjoint. This can be utilized to parallelize or vectorize our algorithm. The correctness is verified by comparing it with the analytical solution based on a three-layered model. The numerical performance is examined by comparing it with other commonly used state-of-art solvers based on three increasingly more complex models, indicating the efficiency dominance, good parallelization, and excellent ability on handling grid stretching for our algorithm.