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Decay rates of adaptive finite elements with Dörfler marking

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Abstract

We investigate the decay rate for an adaptive finite element discretization of a second order linear, symmetric, elliptic PDE. We allow for any kind of estimator that is locally equivalent to the standard residual estimator. This includes in particular hierarchical estimators, estimators based on the solution of local problems, estimators based on local averaging, equilibrated residual estimators, the ZZ-estimator, etc. The adaptive method selects elements for refinement with Dörfler marking and performs a minimal refinement in that no interior node property is needed. Based on the local equivalence to the residual estimator we prove an error reduction property. In combination with minimal Dörfler marking this yields an optimal decay rate in terms of degrees of freedom.

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References

  1. Ainsworth M., Oden J.T.: A posteriori error estimation in finite element analysis. Wiley-Interscience, New York (2000)

    MATH  Google Scholar 

  2. Bänsch E.: Local mesh refinement in 2 and 3 dimensions. IMPACT Comput. Sci. Eng. 3, 181–191 (1991)

    Article  MATH  Google Scholar 

  3. Binev P., Dahmen W., DeVore R.: Adaptive finite element methods with convergence rates. Numer. Math. 97, 219–268 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  4. Binev P., Dahmen W., DeVore R., Petrushev P.: Approximation classes for adaptive methods. Serdica Math. J. 28(4), 391–416 (2002) Dedicated to the memory of Vassil Popov on the occasion of his 60th birthday. MRMR1965238 (2004b:65176)

    MathSciNet  MATH  Google Scholar 

  5. Bornemann F.A., Erdmann B., Kornhuber R.: A posteriori error estimates for elliptic problems in two and three space dimensions. SIAM J. Numer. Anal. 33(3), 1188–1204 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  6. Braess D., Pillwein V., Schöberl J.: Equilibrated residual error estimates are p-robust. Comput. Methods Appl. Mech. Eng. 198(13–14), 1189–1197 (2009)

    Article  MATH  Google Scholar 

  7. Bangerth, W., Rannacher, R.: Adaptive finite element methods for differential equations. In: Lectures in Mathematics, ETH Zürich. Birkhäuser, Basel (2003)

  8. Braess, D.: Finite Elements, 3rd edn. Cambridge University Press, Cambridge (2007). Theory, fast solvers, and applications in elasticity theory. Translated from German by Larry L. Schumaker

  9. Babuška, I., Strouboulis, T.: The finite element method and its reliability. In: Numerical Mathematics and Scientific Computation. The Clarendon Press, New York (2001)

  10. Braess D., Schöberl J.: Equilibrated residual error estimator for edge elements. Math. Comput. 77(262), 651–672 (2008)

    MATH  Google Scholar 

  11. Bernardi C., Verfürth R.: Adaptive finite element methods for elliptic equations with non-smooth coefficients. Numer. Math. 85(4), 579–608 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Carstensen C., Bartels S.: Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part I: low order conforming, nonconforming, and mixed FEM. Math. Comput. 71, 945–969 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Chen Z., Feng J.: An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems. Math. Comput. 73, 1167–1193 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Cascón J.M., Kreuzer C., Nochetto R.H., Siebert K.G.: Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46(5), 2524–2550 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Clément P.: Approximation by finite element functions using local regularization. RAIRO 9, 77–84 (1975)

    Google Scholar 

  16. Diening L., Kreuzer Ch.: Convergence of an adaptive finite element method for the p-Laplacian equation. SIAM J. Numer. Anal. 46(2), 614–638 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Dörfler W., Nochetto R.H.: Small data oscillation implies the saturation assumption. Numer. Math. 91(1), 1–12 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Dörfler W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33, 1106–1124 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  19. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. In: Classics in Mathematics. Springer, New York (2001)

  20. Kossaczký I.: A recursive approach to local mesh refinement in two and three dimensions. J. Comput. Appl. Math. 55, 275–288 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  21. Maubach J.M.: Local bisection refinement for n-simplicial grids generated by reflection. SIAM J. Sci. Comput. 16, 210–227 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  22. Mekchay K., Nochetto R.H.: Convergence of adaptive finite element methods for general second order linear elliptic PDEs. SIAM J. Numer. Anal. 43(5), 1803–1827 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Morin P., Nochetto R.H., Siebert K.G.: Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38, 466–488 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  24. Morin P., Nochetto R.H., Siebert K.G.: Convergence of adaptive finite element methods. SIAM Rev. 44, 631–658 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  25. Morin P., Nochetto R.H., Siebert K.G.: Local problems on stars: a posteriori error estimators, convergence, and performance. Math. Comput. 72, 1067–1097 (2003)

    MathSciNet  MATH  Google Scholar 

  26. Morin P., Siebert K.G., Veeser A.: A basic convergence result for conforming adaptive finite elements. Math. Models Methods Appl. 18, 707–737 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. Nochetto R.H., Siebert K.G., Veeser A.: Theory of adaptive finite element methods: an introduction. In: DeVore, R.A., Kunoth, A. (eds) Multiscale, Nonlinear and Adaptive Approximation, pp. 409–542. Springer, New York (2009)

    Chapter  Google Scholar 

  28. Petzoldt, M.: Regularity and error estimators for elliptic equations with discontinuous coefficients. Ph.D. thesis, FU Berlin (2001)

  29. Petzoldt M.: A posteriori error estimators for elliptic equations with discontinuous coefficients. Adv. Comput. Math. 16, 47–75 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  30. Prager W., Synge J.L.: Approximations in elasticity based on the concept of function space. Quart. Appl. Math. 5, 241–269 (1947)

    MathSciNet  MATH  Google Scholar 

  31. Siebert, K.G.: A convergence proof for adaptive finite elements without lower bound. IMA J. Numer. Anal. doi:10.1093/imanum/drq001 (published online) (2010, May)

  32. Schmidt, A., Siebert, K.G.: Design of adaptive finite element software. The finite element toolbox ALBERTA. In: Lecture Notes in Computational Science and Engineering, vol. 42. Springer, New York (2005)

  33. Stevenson R.: Optimality of a standard adaptive finite element method. Found. Comput. Math. 7(2), 245–269 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  34. Stevenson R.: The completion of locally refined simplicial partitions created by bisection. Math. Comput. 77(261), 227–241 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  35. Siebert K.G., Veeser A.: A unilaterally constrained quadratic minimization with adaptive finite elements. SIAM J. Optim. 18(1), 260–289 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  36. Scott L.R., Zhang S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54(190), 483–493 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  37. Traxler C.T.: An algorithm for adaptive mesh refinement in n dimensions. Computing 59, 115–137 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  38. Veeser A.: Convergent adaptive finite elements for the nonlinear Laplacian. Numer. Math. 92(4), 743–770 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  39. Verfürth, R.: A review of a posteriori error estimation and adaptive mesh-refinement techniques. Adv. Numer. Math. John Wiley, Chichester (1996)

  40. Verfürth R.: Robust a posteriori error estimators for a singularly perturbed reaction-diffusion equation. Numer. Math. 78(3), 479–493 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  41. Veeser, A., Verfürth, R.: Explicit upper bounds for dual norms of residuals. SIAM J. Numer. Anal. (2009, to appear)

  42. Zienkiewicz O.C., Zhu J.Z.: A simple error estimator and adaptive procedure for practical engineering analysis. Int. J. Numer. Methods Eng. 24, 337–357 (1987)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Fakultät für Mathematik, Universität Duisburg-Essen, Forsthausweg 2, 47057, Duisburg, Germany

    Christian Kreuzer & Kunibert G. Siebert

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  1. Christian Kreuzer
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  2. Kunibert G. Siebert
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Correspondence to Christian Kreuzer.

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Kreuzer, C., Siebert, K.G. Decay rates of adaptive finite elements with Dörfler marking. Numer. Math. 117, 679–716 (2011). https://doi.org/10.1007/s00211-010-0324-5

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  • DOI: https://doi.org/10.1007/s00211-010-0324-5

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