Method for Analytically Finding the Nullspace of Discretized Curl–Curl Operator in Unsymmetrical Finite-Difference Subgridding Algorithms

Subgridding schemes enable efficient electromagnetic simulations by refining the mesh only in regions requiring higher resolution. However, like conformal-mesh-based methods, they suffer from low-frequency and dense-mesh breakdown. The solution to this problem lies in the nullspace of the curl–curl operator. In this article, we develop an analytical approach to constructing the nullspace of the discretized curl–curl operator in finite-difference subgridding methods, eliminating the need for computation. Due to the asymmetry often introduced to meet accuracy requirements, subgridding methods yield distinct left and right nullspaces. Our proposed method effectively determines both. The analytically derived nullspaces are then applied to solve low-frequency breakdown and develop fast and robust broadband solutions of Maxwell’s equations in multiscale inhomogeneous problems. Numerical experiments demonstrate the accuracy and efficiency of our approach.