Collective marking for arbitrary order adaptive least-squares finite element methods with optimal rates

Abstract

The collective marking strategy with alternative refinement-indicators in adaptive mesh-refining of least-squares finite element methods (LSFEMs) has recently been shown to lead to optimal convergence rates in Carstensen (2020). The proofs utilize explicit identities for the lowest-order Raviart–Thomas and the Crouzeix–Raviart finite elements. This paper generalizes those results to arbitrary polynomial degree and mixed boundary conditions with some novel arguments. The analysis is outlined for the Poisson equation in 3D with mixed boundary conditions.