A Broadband Preconditioner Based on Sparsified Nested Dissection Ordering Technique for the Vector-Scalar Potential Discrete Exterior Calculus Solver

The discrete exterior calculus (DEC) $\mathbf {A}$-$\Phi$ solver is a broadband stable solver in computational electromagnetics which can work from DC to optics. In order to solve practical problems, which are often multi-scale ones with large number of unknowns and condition number, a broadband preconditioner to the DEC $\mathbf {A}$-$\Phi$ solver is proposed in this paper. The proposed preconditioner is based on sparsified nested dissection ordering (spa-NDO) technique. In this paper, introductions to the DEC $\mathbf {A}$-$\Phi$ solver and NDO technique are provided, as well as detailed implementation flow of the proposed modified spa-NDO preconditioner. Through numerical examples, it reveals that the proposed preconditioned solver has $O(N \log N)$ computational complexity. The efficiency of the proposed preconditioner is almost independent of parameters such as frequency and conductivity in the problem, which indicates its broadband nature.