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An effective method is developed in this work to extend the validity of a full-wave finite-element-based solution down to dc for general 3-D problems. In this method, we accurately decompose the Maxwell's system at low frequencies into two subsystems in the framework of a full-wave-based solution. One has an analytical frequency dependence, whereas the other can be solved at frequencies as low as dc. Thus, we bypass the numerical difficulty of solving a highly ill-conditioned and even singular system at low frequencies. In addition, we provide a theoretical analysis on the conditioning of the matrices of the original coupled Maxwell's system and the decomposed system. We show that the decomposed system is well conditioned, and also positive definite at dc. The validity and accuracy of the proposed method have been demonstrated by extraction of state-of-the-art on-chip integrated circuits at frequencies as low as dc.