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The tree-cotree splitting algorithm is applied to the time-domain finite-element analysis of electromagnetic problems to accelerate the time-marching process. Based on the splitting of finite-element edges into tree and cotree edges, two sets of basis functions are formed to expand the electric field and construct the time-marching system. Inclusion of the time-domain waveguide port boundary condition and lumped port interfaces is also discussed. It is shown that application of the tree-cotree splitting to the time-domain finite-element analysis maintains the solution accuracy while significantly reducing the iteration count per time step for a preconditioned iterative solution when the time-step size becomes relatively large. As a result, it allows adoption of a larger time-step size within the requirement of the temporal sampling rate to achieve a faster time-marching process with a marginal additional cost. The tree-cotree splitting algorithm also suppresses the late-time linear drift associated with the conventional time-domain finite-element method.