Convergence and Optimality of hp-AFEM
This work provides the first convergence and optimality proof for an hp-adaptive FEM, addressing a long-standing open problem for practitioners and theorists in numerical analysis.
The authors design an hp-adaptive finite element method (hp-AFEM) for dimensions 1 and 2, proving its convergence and instance optimality. The algorithm iterates between a coarsening step (hp-NEARBEST) and a refinement step (REDUCE).
We design and analyze an adaptive $hp$-finite element method (hp-AFEM) in dimensions $n=1,2$. The algorithm consists of iterating two routines: hp-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.