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We develop an adaptive spectral element method to price American options, whose solutions contain a moving singularity, automatically and to within prescribed errors. The adaptive algorithm uses an error estimator to determine where refinement or de-refinement is needed and a work estimator to decide whether to change the element size or the polynomial order. We derive two local error estimators and a global error estimator. The local error estimators are derived from the Legendre coefficients and the global error estimator is based on the adjoint problem. One local error estimator uses the rate of decay of the Legendre coefficients to estimate the error. The other local error estimator compares the solution to an estimated solution using fewer Legendre coefficients found by the Tau method. The global error estimator solves the adjoint problem to weight local error estimates to approximate a terminal error functional. Both types of error estimators produce meshes that match expectations by being fine near the early exercise boundary and strike price and coarse elsewhere. The produced meshes also adapt as expected by de-refining near the strike price as the solution smooths and staying fine near the moving early exercise boundary. Both types of error estimators also give solutions whose error is within prescribed tolerances. The adjoint-based error estimator is more flexible, but costs up to three times as much as using the local error estimate alone. The global error estimator has the advantages of tracking the accumulation of error in time and being able to discount large local errors that do not affect the chosen terminal error functional. The local error estimator is cheaper to compute because the global error estimator has the added cost of solving the adjoint problem.