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

M. Aurada; M. Feischl; J. Kemetmüller; M. Page; D. Praetorius

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2013)

  • Volume: 47, Issue: 4, page 1207-1235
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Aurada, M., et al. "Each H1/2–stable projection yields convergence and quasi–optimality of adaptive FEM with inhomogeneous Dirichlet data in Rd." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 47.4 (2013): 1207-1235. <http://eudml.org/doc/273190>.

@article{Aurada2013,
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.},
author = {Aurada, M., Feischl, M., Kemetmüller, J., Page, M., Praetorius, D.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {adaptive finite element method; convergence analysis; quasi–optimality; inhomogeneous Dirichlet data; convergence; quasi-optimality; stability; second-order elliptic equations; Poisson equation; mixed Dirichlet-Neumann boundary conditions; Scott-Zhang projection; error estimation; numerical experiments},
language = {eng},
number = {4},
pages = {1207-1235},
publisher = {EDP-Sciences},
title = {Each H1/2–stable projection yields convergence and quasi–optimality of adaptive FEM with inhomogeneous Dirichlet data in Rd},
url = {http://eudml.org/doc/273190},
volume = {47},
year = {2013},
}

TY - JOUR
AU - Aurada, M.
AU - Feischl, M.
AU - Kemetmüller, J.
AU - Page, M.
AU - Praetorius, D.
TI - Each H1/2–stable projection yields convergence and quasi–optimality of adaptive FEM with inhomogeneous Dirichlet data in Rd
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2013
PB - EDP-Sciences
VL - 47
IS - 4
SP - 1207
EP - 1235
AB - 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.
LA - eng
KW - adaptive finite element method; convergence analysis; quasi–optimality; inhomogeneous Dirichlet data; convergence; quasi-optimality; stability; second-order elliptic equations; Poisson equation; mixed Dirichlet-Neumann boundary conditions; Scott-Zhang projection; error estimation; numerical experiments
UR - http://eudml.org/doc/273190
ER -

References

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  1. [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). Zbl1275.65078MR3082295
  2. [2] M. Aurada, S. Ferraz-Leite and D. Praetorius, Estimator reduction and convergence of adaptive BEM. Appl. Numer. Math. 62 (2012). Zbl1237.65131MR2908795
  3. [3] M. Ainsworth and T. Oden, A posteriori error estimation in finite element analysis, Wiley–Interscience, New-York (2000). Zbl1008.65076MR1885308
  4. [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. Zbl1062.65113MR2101782
  5. [5] P. Binev, W. Dahmen and R. DeVore, Adaptive finite element methods with convergence rates, Numer. Math.97 (2004) 219–268. Zbl1063.65120MR2050077
  6. [6] P. Binev, W. Dahmen, R. DeVore and P. Petrushev, Approximation Classes for Adaptive Methods. Serdica. Math. J.28 (2002) 391–416. Zbl1039.42030MR1965238
  7. [7] R. Becker and S. Mao, Convergence and quasi–optimal complexity of a simple adaptive finite element method. ESAIM: M2AN 43 (2009) 1203–1219. Zbl1179.65139MR2588438
  8. [8] I. Babuška and M. Vogelius, Feedback and adaptive finite element solution of one-dimensional boundary value problems. Numer. Math.44 (1984) 75–102. Zbl0574.65098MR745088
  9. [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. Zbl1018.65138MR1872725
  10. [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. Zbl1176.65122MR2421046
  11. [11] M. Cascón, R. Nochetto: Quasioptimal cardinality of AFEM driven by nonresidual estimators. IMA J. Numer. Anal.32 (2012) 1–29. Zbl1242.65237MR2875241
  12. [12] W. Dörfler: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal.33 (1996) 1106–1124. Zbl0854.65090MR1393904
  13. [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). Zbl1273.65186MR3047442
  14. [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). Zbl1291.65341
  15. [15] F. Gaspoz and P. Morin, Approximation classes for adaptive higher order finite element approximation. To appear in Math. Comput. (2012). Zbl1298.41024MR3223327
  16. [16] George C. Hsiao, Wolfgang and L. Wendland, Boundary Integral Equations. Springer Verlag, Berlin (2008). Zbl1157.65066MR2441884
  17. [17] C. Kreuzer and K. Siebert, Decay rates of adaptive finite elements with Dörfler marking. Numer. Math.117 (2011) 679–716. Zbl1219.65133MR2776915
  18. [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). Zbl1280.65116
  19. [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). Zbl1302.65267MR3097045
  20. [20] W. McLean, Strongly elliptic systems and boundary integral equations. Cambridge University Press, Cambridge (2000). Zbl0948.35001MR1742312
  21. [21] P. Morin, R. Nochetto and K. Siebert, Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal.18 (2000) 466–488. Zbl0970.65113MR1770058
  22. [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. Zbl1019.65083MR1972728
  23. [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. Zbl1153.65111MR2413035
  24. [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. Zbl1092.65098MR2238754
  25. [25] S. Sauter and C. Schwab, Randelementmethoden. Springer, Wiesbaden (2004). 
  26. [26] L. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput54 (1990) 483–493. Zbl0696.65007MR1011446
  27. [27] R. Stevenson: Optimality of standard adaptive finite element method. Found. Comput. Math. (2007) 245–269. Zbl1136.65109MR2324418
  28. [28] R. Stevenson, The completion of locally refined simplicial partitions created by bisection. Math. Comput.77 (2008) 227–241. Zbl1131.65095MR2353951
  29. [29] Traxler: An Algorithm for Adaptive Mesh Refinement in n Dimensions. Computing59 (1997) 115–137. Zbl0944.65123MR1475530
  30. [30] R. Verfürth, A review of a posteriori error estimation and adaptive mesh–refinement techniques. Wiley–Teubner (1996). Zbl0853.65108

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