Convergence and optimality of hp-AFEM

We design and analyze an adaptive hp-finite element method ((Formula presented.)-AFEM) in dimensions (Formula presented.). The algorithm consists of iterating two routines: (Formula presented.)-NEARBEST finds a near-best hp-approximation of the current discrete solution and data to a desired accuracy, and REDUCE improves the discrete solution to a finer but comparable accuracy. The former hinges on a recent algorithm by Binev for adaptive hp-approximation, and acts as a coarsening step. We prove convergence and instance optimality.