Each H^1/2–stable projection yields convergence and quasi–optimality of adaptive FEM with inhomogeneous Dirichlet data in R^d

Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8-10, 1040 Wien, Austria.
Markus.Aurada@tuwien.ac.at, Michael.Feischl@tuwien.ac.at, Josef.Kemetmueller@tuwien.ac.at, Dirk.Praetorius@tuwien.ac.at; Marcus.Page@tuwien.ac.at

Received: 23 February 2012
Revised: 22 September 2012

Abstract

We consider the solution of second order elliptic PDEs in R^d with inhomogeneous Dirichlet data by means of an h–adaptive FEM with fixed polynomial order p ∈ N. As model example serves the Poisson equation with mixed Dirichlet–Neumann boundary conditions, where the inhomogeneous Dirichlet data are discretized by use of an H^1 / 2–stable projection, for instance, the L²–projection for p = 1 or the Scott–Zhang projection for general p ≥ 1. For error estimation, we use a residual error estimator which includes the Dirichlet data oscillations. We prove that each H^1 / 2–stable projection yields convergence of the adaptive algorithm even with quasi–optimal convergence rate. Numerical experiments with the Scott–Zhang projection conclude the work.