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}, publisher = {EDP-Sciences}, volume = {47}, number = {4}, year = {2013}, doi = {10.1051/m2an/2013069}, zbl = {1275.65078}, language = {en}, url = {http://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 - http://www.numdam.org/articles/10.1051/m2an/2013069/ DO - 10.1051/m2an/2013069 LA - en ID - M2AN_2013__47_4_1207_0 ER -
%0 Journal Article %A Aurada, M. %A Feischl, M. %A Kemetmüller, J. %A Page, M. %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 http://www.numdam.org/articles/10.1051/m2an/2013069/ %R 10.1051/m2an/2013069 %G en %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 , Volume 47 (2013) no. 4, pp. 1207-1235. doi : 10.1051/m2an/2013069. http://www.numdam.org/articles/10.1051/m2an/2013069/
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