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Each H1/2–stable projection yields convergence and quasi–optimality of adaptive FEM with inhomogeneous Dirichlet data in 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 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 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

M. Aurada, M. Feischl, J. Kemetmüller, M. Page and D. Praetorius, Each H 1 / 2–stable projection yields convergence and quasi–optimality of adaptive FEM with inhomogeneous Dirichlet data in Rd(extended preprint) ASC Report 03/2012, Institute for Analysis and Scientific Computing, Vienna University of Technology (2012).
Aurada, M., Ferraz-Leite, S. and Praetorius, D., Estimator reduction and convergence of adaptive BEM. Appl. Numer. Math. 62 (2012). Google ScholarPubMed
M. Ainsworth and T. Oden, A posteriori error estimation in finite element analysis, Wiley–Interscience, New-York (2000).
Bartels, S., Carstensen, C. and Dolzmann, G., Inhomogeneous Dirichlet conditions in a priori and a posteriori finite element error analysis. Numer. Math. 99 (2004) 1–24. Google Scholar
Binev, P., Dahmen, W. and DeVore, R., Adaptive finite element methods with convergence rates, Numer. Math. 97 (2004) 219–268. Google Scholar
Binev, P., Dahmen, W., DeVore, R. and Petrushev, P., Approximation Classes for Adaptive Methods. Serdica. Math. J. 28 (2002) 391–416. Google Scholar
Becker, R. and Mao, S., Convergence and quasi–optimal complexity of a simple adaptive finite element method. ESAIM: M2AN 43 (2009) 1203–1219. Google Scholar
Babuška, I. and Vogelius, M., Feedback and adaptive finite element solution of one-dimensional boundary value problems. Numer. Math. 44 (1984) 75–102. Google Scholar
Carstensen, C., Maischak, M. and Stephan, E.P., A posteriori error estimate and h-adaptive algorithm on surfaces for Symm’s integral equation. Numer. Math. 90 (2001) 197–213. Google Scholar
Cascón, M., Kreuzer, C., Nochetto, R. and Siebert, K.: quasi–optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46 (2008) 2524–2550. Google Scholar
Cascón, M., Nochetto, R.: Quasioptimal cardinality of AFEM driven by nonresidual estimators. IMA J. Numer. Anal. 32 (2012) 1–29. Google Scholar
Dörfler, W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33 (1996) 1106–1124. Google Scholar
M. Feischl, M. Karkulik, M. Melenk and D. Praetorius, Quasi–optimal convergence rate for an adaptive boundary element method. SIAM J. Numer. Anal. (2013).
M. Feischl, M. Page and D. Praetorius, Convergence and quasi–optimality of adaptive FEM with inhomogeneous Dirichlet data, ASC Report 34/2010, Institute for Analysis and Scientific Computing, Vienna University of Technology (2010).
F. Gaspoz and P. Morin, Approximation classes for adaptive higher order finite element approximation. To appear in Math. Comput. (2012).
George C. Hsiao, Wolfgang and L. Wendland, Boundary Integral Equations. Springer Verlag, Berlin (2008).
Kreuzer, C. and Siebert, K., Decay rates of adaptive finite elements with Dörfler marking. Numer. Math. 117 (2011) 679–716. Google Scholar
M. Karkulik, G. Of and D. Praetorius, Convergence of adaptive 3D BEM for some weakly singular integral equations based on isotropic mesh–refinement. Numer. Methods Partial Differ. Eq. (2013).
M. Karkulik, D. Pavlicek and D. Praetorius, On 2D newest vertex bisection: Optimality of mesh-closure and H 1–stability of L 2–projection. Constr. Approx. (2013).
W. McLean, Strongly elliptic systems and boundary integral equations. Cambridge University Press, Cambridge (2000).
Morin, P., Nochetto, R. and Siebert, K., Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 18 (2000) 466–488. Google Scholar
Morin, P., Nochetto, R. and Siebert, K., Local problems on stars: a posteriori error estimators, convergence, and performance. Math. Comput. 72 (2003) 1067–1097. Google Scholar
Morin, P., Siebert, K. and Veeser, A., A basic convergence result for conforming adaptive finite elements. Math. Models Methods Appl. Sci. 18 (2008) 707–737. Google Scholar
Sacchi, R. and Veeser, A., Locally efficient and reliable a posteriori error estimators for Dirichlet problems. Math. Models Methods Appl. Sci. 16 (2006) 319–346. Google Scholar
S. Sauter and C. Schwab, Randelementmethoden. Springer, Wiesbaden (2004).
Scott, L. and Zhang, S., Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput 54 (1990) 483–493. Google Scholar
R. Stevenson: Optimality of standard adaptive finite element method. Found. Comput. Math. (2007) 245–269.
Stevenson, R., The completion of locally refined simplicial partitions created by bisection. Math. Comput. 77 (2008) 227–241. Google Scholar
Traxler: An Algorithm for Adaptive Mesh Refinement in n Dimensions. Computing 59 (1997) 115–137.
R. Verfürth, A review of a posteriori error estimation and adaptive mesh–refinement techniques. Wiley–Teubner (1996).