Each H1/2–stable projection yields convergence and quasi–optimality of adaptive FEM with inhomogeneous Dirichlet data in Rd
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
Revised: 22 September 2012
We consider the solution of second order elliptic PDEs in Rd 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 H1 / 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 H1 / 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.
Mathematics Subject Classification: 65N30 / 65N50
Key words: Adaptive finite element method / convergence analysis / quasi–optimality / inhomogeneous Dirichlet data
© EDP Sciences, SMAI, 2013