Efficient and Exact Method for Correcting the Ill-Posedness of the Discrete Maxwell System

The numerical system resulting from a partial differential equation based solution of Maxwell's equations is ill-posed due to the rank-deficient curl-operator. We find an efficient and exact method to perform rank-revealing row reduction of the discretized curl-operator to zero out the redundant rows during the numerical discretization. The method is analytical, and its computational cost is trivial. Using the resulting full-rank curl-operator, we further transform the original ill-posed system into a well-posed one whose solution is stable at all frequencies and scales, while simultaneously accelerating the solution of the numerical Maxwell system. Applications to general 3-D electromagnetic analysis have demonstrated its validity.