An Adaptive hp–DG–FE Method for Elliptic Problems: Convergence and Optimality in the 1D Case

We propose and analyze an hp-adaptive DG–FEM algorithm, termed hp-ADFEM, and its one-dimensional realization, which is convergent, instance optimal, and h- and p-robust. The procedure consists of iterating two routines: one hinges on Binev’s algorithm for the adaptive hp-approximation of a given function, and finds a near-best hp-approximation of the current discrete solution and data to a desired accuracy; the other one improves the dis- crete solution to a finer but comparable accuracy, by iteratively applying Dörfler marking and h refinement.