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Each H1/2–stable projectionyields convergence and quasi–optimality of adaptive FEM with inhomogeneous Dirichlet datain Rd

Published online by Cambridge University Press:  17 June 2013

M. Aurada ,
M. Feischl ,
J. Kemetmüller ,
M. Page  and
D. Praetorius
M. Aurada
Affiliation:
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
M. Feischl
Affiliation:
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
J. Kemetmüller
Affiliation:
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
M. Page
Affiliation:
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
D. Praetorius
Affiliation:
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
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Abstract

We consider the solution of second order elliptic PDEs in Rdwith inhomogeneous Dirichlet data by means of an h–adaptive FEM withfixed polynomial order p ∈ N. As model example serves the Poissonequation with mixed Dirichlet–Neumann boundary conditions, where the inhomogeneousDirichlet data are discretized by use of an H1 / 2–stableprojection, for instance, the L2–projection forp = 1 or the Scott–Zhang projection for general p ≥ 1.For error estimation, we use a residual error estimator which includes the Dirichlet dataoscillations. We prove that each H1 / 2–stable projectionyields 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 methodconvergence analysisquasi–optimalityinhomogeneous Dirichlet data
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 4: Direct and inverse modeling of the cardiovascular and respiratory systems , July 2013 , pp. 1207 - 1235
Copyright
© EDP Sciences, SMAI, 2013

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References

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