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An unconditionally stable and computationally efficient time-domain finite-element method is developed to solve large-scale IC and package problems. In this method, an analytical expression of the time dependence is developed for the field unknowns inside conductors. And hence any time step can be used to stably solve the system of equations inside conductors. A matrix solution is involved in the analytical expression. It is efficiently obtained by the time-domain finite-element reduction-recovery method, the factorization cost of which is O(M), with M much less than the system matrix size N. The system of equations exterior to conductors is formed by a backward difference method, and hence is also unconditionally stable. The resultant system matrix is solved efficiently by an H-matrix based direct sparse solver, which is shown to outperform the state-of-the-art direct sparse solver. The system exterior to the conductors and that interior to the conductors are then solved by a staggered marching scheme, the convergence of which is theoretically proved. Applications to on-chip problems have demonstrated a time step that is three orders of magnitude larger than what is permitted by an explicit time-domain scheme, with fast CPU run time, modest memory consumption, and without sacrificing accuracy.