Fast Method for Accelerating Time-Domain Solutions of Ill-Conditioned Electromagnetic Problems by Changing Curl–Curl Operator to Laplacian

The ill-conditioned numerical system arising from an electromagnetic analysis is often due to the discretized curl–curl operator which has both zero eigenvalues and large ones inversely proportional to the square of the smallest feature size. In this work, we analytically decompose the discretized curl–curl operator into a gradient divergence operator and a Laplacian without numerical computation. We further split the field solution into a gradient field (curl-free component) and a divergence-free one to take advantage of the decomposition. The gradient divergence operator vanishes when acting on the divergence-free component of the field solution. As a result, we can replace the highly ill-conditioned curl–curl operator with a Laplacian for computing the divergence-free component regardless of inhomogeneity and source settings. Since the Laplacian is positive definite and well-conditioned, the new system matrix can converge quickly in a small number of iterations. The method is applied to accelerate time-domain solutions of Maxwell’s equations in ill-conditioned problems such as integrated circuits (ICs). Numerical results have validated its accuracy and efficiency.