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 , Tome 47 (2013) no. 4, pp. 1207-1235.

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.

DOI : 10.1051/m2an/2013069
Classification : 65N30, 65N50
Mots clés : 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 , Tome 47 (2013) no. 4, pp. 1207-1235. doi : 10.1051/m2an/2013069. http://www.numdam.org/articles/10.1051/m2an/2013069/

[1] M. Aurada, M. Feischl, J. Kemetmüller, M. Page and D. Praetorius, Each H1 / 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). | MR

[2] M. Aurada, S. Ferraz-Leite and D. Praetorius, Estimator reduction and convergence of adaptive BEM. Appl. Numer. Math. 62 (2012). | MR | Zbl

[3] M. Ainsworth and T. Oden, A posteriori error estimation in finite element analysis, Wiley-Interscience, New-York (2000). | MR | Zbl

[4] S. Bartels, C. Carstensen and G. Dolzmann, Inhomogeneous Dirichlet conditions in a priori and a posteriori finite element error analysis. Numer. Math. 99 (2004) 1-24. | MR | Zbl

[5] P. Binev, W. Dahmen and R. Devore, Adaptive finite element methods with convergence rates, Numer. Math. 97 (2004) 219-268. | MR | Zbl

[6] P. Binev, W. Dahmen, R. Devore and P. Petrushev, Approximation Classes for Adaptive Methods. Serdica. Math. J. 28 (2002) 391-416. | MR | Zbl

[7] R. Becker and S. Mao, Convergence and quasi-optimal complexity of a simple adaptive finite element method. ESAIM: M2AN 43 (2009) 1203-1219. | Numdam | MR | Zbl

[8] I. Babuška and M. Vogelius, Feedback and adaptive finite element solution of one-dimensional boundary value problems. Numer. Math. 44 (1984) 75-102. | MR | Zbl

[9] C. Carstensen, M. Maischak and E.P. Stephan, A posteriori error estimate and h-adaptive algorithm on surfaces for Symm's integral equation. Numer. Math. 90 (2001) 197-213. | MR | Zbl

[10] M. Cascón, C. Kreuzer, R. Nochetto and K. Siebert: quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46 (2008) 2524-2550. | MR | Zbl

[11] M. Cascón, R. Nochetto: Quasioptimal cardinality of AFEM driven by nonresidual estimators. IMA J. Numer. Anal. 32 (2012) 1-29. | MR | Zbl

[12] W. Dörfler: A convergent adaptive algorithm for Poisson's equation. SIAM J. Numer. Anal. 33 (1996) 1106-1124. | MR | Zbl

[13] M. Feischl, M. Karkulik, M. Melenk and D. Praetorius, Quasi-optimal convergence rate for an adaptive boundary element method. SIAM J. Numer. Anal. (2013). | MR | Zbl

[14] 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). | Zbl

[15] F. Gaspoz and P. Morin, Approximation classes for adaptive higher order finite element approximation. To appear in Math. Comput. (2012). | MR | Zbl

[16] George C. Hsiao, Wolfgang and L. Wendland, Boundary Integral Equations. Springer Verlag, Berlin (2008). | MR | Zbl

[17] C. Kreuzer and K. Siebert, Decay rates of adaptive finite elements with Dörfler marking. Numer. Math. 117 (2011) 679-716. | MR | Zbl

[18] 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). | Zbl

[19] M. Karkulik, D. Pavlicek and D. Praetorius, On 2D newest vertex bisection: Optimality of mesh-closure and H1-stability of L2-projection. Constr. Approx. (2013). | MR

[20] W. Mclean, Strongly elliptic systems and boundary integral equations. Cambridge University Press, Cambridge (2000). | MR | Zbl

[21] P. Morin, R. Nochetto and K. Siebert, Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 18 (2000) 466-488. | MR | Zbl

[22] P. Morin, R. Nochetto and K. Siebert, Local problems on stars: a posteriori error estimators, convergence, and performance. Math. Comput. 72 (2003) 1067-1097. | MR | Zbl

[23] P. Morin, K. Siebert and A. Veeser, A basic convergence result for conforming adaptive finite elements. Math. Models Methods Appl. Sci. 18 (2008) 707-737. | MR | Zbl

[24] R. Sacchi and A. Veeser, Locally efficient and reliable a posteriori error estimators for Dirichlet problems. Math. Models Methods Appl. Sci. 16 (2006) 319-346. | MR | Zbl

[25] S. Sauter and C. Schwab, Randelementmethoden. Springer, Wiesbaden (2004).

[26] L. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput 54 (1990) 483-493. | MR | Zbl

[27] R. Stevenson: Optimality of standard adaptive finite element method. Found. Comput. Math. (2007) 245-269. | MR | Zbl

[28] R. Stevenson, The completion of locally refined simplicial partitions created by bisection. Math. Comput. 77 (2008) 227-241. | MR | Zbl

[29] Traxler: An Algorithm for Adaptive Mesh Refinement in n Dimensions. Computing 59 (1997) 115-137. | MR | Zbl

[30] R. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley-Teubner (1996). | Zbl

Cité par Sources :