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.
Keywords: adaptive finite element method, convergence analysis, quasi-optimality, inhomogeneous Dirichlet data
@article{M2AN_2013__47_4_1207_0,
author = {Aurada, M. and Feischl, M. and Kemetm\"uller, J. and Page, M. and Praetorius, D.},
title = {Each $H^{1/2}$-stable projection yields convergence and quasi-optimality of adaptive {FEM} with inhomogeneous {Dirichlet} data in $R^d$},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1207--1235},
year = {2013},
publisher = {EDP Sciences},
volume = {47},
number = {4},
doi = {10.1051/m2an/2013069},
zbl = {1275.65078},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2013069/}
}
TY - JOUR
AU - Aurada, M.
AU - Feischl, M.
AU - Kemetmüller, J.
AU - Page, M.
AU - Praetorius, D.
TI - Each $H^{1/2}$-stable projection yields convergence and quasi-optimality of adaptive FEM with inhomogeneous Dirichlet data in $R^d$
JO - ESAIM: Mathematical Modelling and Numerical Analysis
PY - 2013
SP - 1207
EP - 1235
VL - 47
IS - 4
PB - EDP Sciences
UR - https://www.numdam.org/articles/10.1051/m2an/2013069/
DO - 10.1051/m2an/2013069
LA - en
ID - M2AN_2013__47_4_1207_0
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%A Praetorius, D.
%T Each $H^{1/2}$-stable projection yields convergence and quasi-optimality of adaptive FEM with inhomogeneous Dirichlet data in $R^d$
%J ESAIM: Mathematical Modelling and Numerical Analysis
%D 2013
%P 1207-1235
%V 47
%N 4
%I EDP Sciences
%U https://www.numdam.org/articles/10.1051/m2an/2013069/
%R 10.1051/m2an/2013069
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%F M2AN_2013__47_4_1207_0
Aurada, M.; Feischl, M.; Kemetmüller, J.; Page, M.; Praetorius, D. Each $H^{1/2}$-stable projection yields convergence and quasi-optimality of adaptive FEM with inhomogeneous Dirichlet data in $R^d$. ESAIM: Mathematical Modelling and Numerical Analysis , Direct and inverse modeling of the cardiovascular and respiratory systems. Numéro spécial, Tome 47 (2013) no. 4, pp. 1207-1235. doi: 10.1051/m2an/2013069
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