Advances in Computational Mathematics

, Volume 27, Issue 1, pp 27–63 | Cite as

Adaptive frame methods for elliptic operator equations

  • Stephan Dahlke
  • Massimo Fornasier
  • Thorsten Raasch
Article

Abstract

This paper is concerned with the development of adaptive numerical methods for elliptic operator equations. We are especially interested in discretization schemes based on frames. The central objective is to derive an adaptive frame algorithm which is guaranteed to converge for a wide range of cases. As a core ingredient we use the concept of Gelfand frames which induces equivalences between smoothness norms and weighted sequence norms of frame coefficients. It turns out that this Gelfand characteristic of frames is closely related to their localization properties. We also give constructive examples of Gelfand wavelet frames on bounded domains. Finally, an application to the efficient adaptive computation of canonical dual frames is presented.

Keywords

operator equations multiscale methods adaptive algorithms domain decomposition sparse matrices overdetermined systems Banach frames norm equivalences Banach spaces 

Mathematics subject classifications (2000)

41A25 41A46 42C15 42C40 46E35 65F10 65F20 65F50 65N12 65N55 65T60 

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References

  1. [1]
    I. Babuška and W.C. Rheinboldt, A posteriori error estimates for finite element methods, Internat. J. Numer. Math. Engrg. 12 (1978) 1597–1615. CrossRefMATHGoogle Scholar
  2. [2]
    R.E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comput. 44 (1985) 283–301. CrossRefMathSciNetMATHGoogle Scholar
  3. [3]
    A. Barinka, T. Barsch, P. Charton, A. Cohen, S. Dahlke, W. Dahmen and K. Urban, Adaptive wavelet schemes for elliptic problems: Implementation and numerical experiments, SIAM J. Sci. Comput. 23(3) (2001) 910–939. CrossRefMathSciNetMATHGoogle Scholar
  4. [4]
    R. Becker, C. Johnson and R. Rannacher, Adaptive error control for multigrid finite element methods, Computing 55 (1995) 271–288. CrossRefMathSciNetMATHGoogle Scholar
  5. [5]
    J. Bergh and J. Löfström, Interpolation Spaces (Springer-Verlag, Berlin, 1976). MATHGoogle Scholar
  6. [6]
    F. Bornemann, B. Erdmann and R. Kornhuber, A posteriori error estimates for elliptic problems in two and three space dimensions, SIAM J. Numer. Anal. 33 (1996) 1188–1204. CrossRefMathSciNetMATHGoogle Scholar
  7. [7]
    L. Borup, Pseudodifferential operators on α-modulation spaces, J. Funct. Spaces Appl. 2(2) (2004) 107–123. MathSciNetMATHGoogle Scholar
  8. [8]
    L. Borup and M. Nielsen, Nonlinear approximation in α-modulation spaces, Preprint (2003). Google Scholar
  9. [9]
    P.G. Casazza and O. Christensen, Approximation of the inverse frame operator and application to Gabor frames, J. Approx. Theory 130(2) (2000) 338–356. CrossRefMathSciNetGoogle Scholar
  10. [10]
    O. Christensen, Finite-dimensional approximation of the inverse frame operator, J. Fourier Anal. Appl. 6(1) (2000) 79–91. CrossRefMathSciNetMATHGoogle Scholar
  11. [11]
    O. Christensen, An Introduction to Frames and Riesz Bases (Birkhäuser, Basel, 2003). MATHGoogle Scholar
  12. [12]
    O. Christensen and T. Strohmer, The finite section method and problems in frame theory, Preprint (2003). Google Scholar
  13. [13]
    C. Chui and W. Stöckler, Nonstationary tight wavelet frames on bounded intervals, Ergebnisberichte Angewandte Mathematik 230, Universität Dortmund (2003). Google Scholar
  14. [14]
    A. Cohen, W. Dahmen and R. DeVore, Adaptive wavelet methods for elliptic operator equations – Convergence rates, Math. Comp. 70 (2001) 27–75. CrossRefMathSciNetMATHGoogle Scholar
  15. [15]
    A. Cohen, W. Dahmen and R. DeVore, Adaptive methods for nonlinear variational problems, Report 221, IGPM, RWTH Aachen (2002). Google Scholar
  16. [16]
    A. Cohen, W. Dahmen and R. DeVore, Adaptive wavelet methods II: Beyond the elliptic case, Found. Comput. Math. 2(3) (2002) 203–245. CrossRefMathSciNetMATHGoogle Scholar
  17. [17]
    E. Cordero and K. Gröchenig, Localization of frames II, Appl. Comput. Harmon. Anal. 17(1) (2004) 29–47. CrossRefMathSciNetMATHGoogle Scholar
  18. [18]
    S. Dahlke, W. Dahmen, R. Hochmuth and R. Schneider, Stable multiscale bases and local error estimation for elliptic problems, Appl. Numer. Math. 23 (1997) 21–48. CrossRefMathSciNetMATHGoogle Scholar
  19. [19]
    S. Dahlke, W. Dahmen and K. Urban, Adaptive wavelet methods for saddle point problems – Optimal convergence rates, SIAM J. Numer. Anal. 40(4) (2002) 1230–1262. CrossRefMathSciNetMATHGoogle Scholar
  20. [20]
    S. Dahlke, M. Fornasier and T. Raasch, Adaptive frame methods for elliptic operator equations, Bericht 2004-3, FB 12 Mathematik und Informatik, Philipps-Universität Marburg (2004). Google Scholar
  21. [21]
    S. Dahlke, R. Hochmuth and K. Urban, Adaptive wavelet methods for saddle point problems, M2AN Math. Model. Numer. Anal. 34 (2000) 1003–1022. CrossRefMathSciNetMATHGoogle Scholar
  22. [22]
    S. Dahlke, G. Steidl and G. Teschke, Weighted coorbit spaces and Banach frames on homogeneous spaces, J. Fourier Anal. Appl. 10(5) (2004) 507–539. CrossRefMathSciNetMATHGoogle Scholar
  23. [23]
    W. Dahmen, Wavelet and multiscale methods for operator equations, Acta Numerica 6 (1997) 55–228. MathSciNetGoogle Scholar
  24. [24]
    W. Dahmen and A. Kunoth, Multilevel preconditioning, Numer. Math. 63(3) (1992) 315–344. CrossRefMathSciNetMATHGoogle Scholar
  25. [25]
    W. Dahmen, A. Kunoth and K. Urban, Biorthogonal spline–wavelets on the interval – Stability and moment conditions, Appl. Comput. Harmon. Anal. 6 (1999) 132–196. CrossRefMathSciNetMATHGoogle Scholar
  26. [26]
    W. Dahmen, S. Prössdorf and R. Schneider, Multiscale methods for pseudodifferential operators on smooth manifolds in: Proc. of the Internat. Conf. on Wavelets: Theory, Algorithms and Applications, eds. C.K. Chui, L. Montefusco and L. Puccio (Academic Press, New York, 1994) pp. 385–424. Google Scholar
  27. [27]
    W. Dahmen and R. Schneider, Wavelets with complementary boundary conditions – Function spaces on the cube, Result. Math. 34(3/4) (1998) 255–293. MathSciNetMATHGoogle Scholar
  28. [28]
    W. Dahmen and R. Schneider, Composite wavelet bases for operator equations, Math. Comp. 68 (1999) 1533–1567. CrossRefMathSciNetMATHGoogle Scholar
  29. [29]
    W. Dahmen and R. Schneider, Wavelets on manifolds I. Construction and domain decomposition, SIAM J. Math. Anal. 31 (1999) 184–230. CrossRefMathSciNetMATHGoogle Scholar
  30. [30]
    W. Dahmen, J. Vorloeper and K. Urban, Adaptive wavelet methods – basic concepts and applications to the Stokes problem, in: Proc. of the Internat. Conf. of Computational Harmonic Analysis, ed. D.-X. Zhou (World Scientific, Singapore, 2002) pp. 39–80. Google Scholar
  31. [31]
    I. Daubechies, Ten Lectures on Wavelets (SIAM, Philadelphia, PA, 1992). MATHGoogle Scholar
  32. [32]
    R. DeVore, Nonlinear approximation, Acta Numerica 7 (1998) 51–150. MathSciNetCrossRefGoogle Scholar
  33. [33]
    R. DeVore and I. Daubechies, Reconstruction of bandlimited function from very coarsely quantized data: A family of stable sigma–delta modulators of arbitrary order, Ann. of Math. 158(2) (2003) 679–710. MathSciNetMATHCrossRefGoogle Scholar
  34. [34]
    W. Dörfler, A convergent adaptive algorithm for Poisson's equation, SIAM J. Numer. Anal. 33 (1996) 737–785. CrossRefGoogle Scholar
  35. [35]
    H.G. Feichtinger and M. Fornasier, Flexible Gabor-wavelet atomic decompositions for L 2-Sobolev spaces, Ann. Mat. Pura Appl. (2004) to appear. Google Scholar
  36. [36]
    H.G. Feichtinger and K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions I, J. Funct. Anal. 86(2) (1989) 307–340. CrossRefMathSciNetMATHGoogle Scholar
  37. [37]
    H.G. Feichtinger and K. Gröchenig, Iterative reconstruction of multivariate band-limited functions from irregular sampling values, SIAM J. Math. Anal. 23(1) (1992) 244–261. CrossRefMathSciNetMATHGoogle Scholar
  38. [38]
    H.G. Feichtinger and T. Strohmer, eds., Gabor Analysis and Algorithms (Birkhäuser, Basel, 1998). MATHGoogle Scholar
  39. [39]
    H.G. Feichtinger and T. Strohmer, eds., Advances in Gabor Analysis (Birkhäuser, Basel, 2003). MATHGoogle Scholar
  40. [40]
    M. Fornasier, Banach frames for alpha-modulation spaces, arXiv:math.FA/0410549. Google Scholar
  41. [41]
    M. Fornasier, Constructive methods for numerical applications in signal processing and homogenization problems, Ph.D. thesis, University of Padova and University of Vienna (2002). Google Scholar
  42. [42]
    M. Fornasier, Quasi-orthogonal decompositions of structured frames, J. Math. Anal. Appl. 289(1) (2004) 180–199. CrossRefMathSciNetMATHGoogle Scholar
  43. [43]
    M. Fornasier and K. Gröchenig, Intrinsic localization of frames, Constr. Approx. (2005) to appear. Google Scholar
  44. [44]
    M. Frazier and B. Jawerth, A discrete transform and decompositions of distribution spaces, J. Fourier Anal. Appl. 93(1) (1990) 34–170. MathSciNetMATHGoogle Scholar
  45. [45]
    K. Gröchenig, Describing functions: atomic decompositions versus frames, Monatsh. Math. 112(1) (1991) 1–42. CrossRefMathSciNetMATHGoogle Scholar
  46. [46]
    K. Gröchenig, Foundations of Time–Frequency Analysis (Birkhäuser, Basel, 2000). Google Scholar
  47. [47]
    K. Gröchenig, Localization of frames, in: GROUP 24: Physical and Mathematical Aspects of Symmetries (Bristol), eds. J.-P. Gazeau, R. Kerner, J.-P. Antoine, S. Metens and J.-Y. Thibon (IOP Publishing, 2003) to appear. Google Scholar
  48. [48]
    K. Gröchenig, Localization of frames, Banach frames, and the invertibility of the frame operator, J. Fourier Anal. Appl. 10(2) (2004) 105–132. CrossRefMathSciNetMATHGoogle Scholar
  49. [49]
    K. Gröchenig and M. Leinert, Symmetry of matrix algebras and symbolic calculus for infinite matrices, Preprint (2003). Google Scholar
  50. [50]
    W. Hackbusch, Elliptic Differential Equations (Springer, Berlin, 1992). MATHGoogle Scholar
  51. [51]
    R.B. Lehoucq and D.C. Sorensen, Deflation techniques for an implicitly restarted Arnoldi iteration, SIAM J. Matrix Anal. Appl. 17(4) (1996) 789–821. CrossRefMathSciNetMATHGoogle Scholar
  52. [52]
    P.G. Lemarié, Bases d'ondelettes sur les groupes de lie stratifiés, Bull. Soc. Math. France 117(2) (1989) 213–232. Google Scholar
  53. [53]
    M. Mommer, Fictitious domain Lagrange multiplier approach: Smoothness analysis, Report 230, IGPM, RWTH Aachen (2003). Google Scholar
  54. [54]
    R. Stevenson, Adaptive solution of operator equations using wavelet frames, SIAM J. Numer. Anal. 41(3) (2003) 1074–1100. CrossRefMathSciNetMATHGoogle Scholar
  55. [55]
    R. Verfürth, A posteriori error estimation and adaptive mesh-refinement techniques, J. Comput. Appl. Math. 50 (1994) 67–83. CrossRefMathSciNetMATHGoogle Scholar
  56. [56]
    M. Werner, Adaptive Frame-Verfahren für elliptische Randwertprobleme, Master thesis, Fachbereich Mathematik und Informatik, Philipps-Universität Marburg (2005) in preparation. Google Scholar
  57. [57]
    A. Zaanen, Integration (North-Holland, Amsterdam, 1967). MATHGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  • Stephan Dahlke
    • 1
  • Massimo Fornasier
    • 2
  • Thorsten Raasch
    • 1
  1. 1.FB 12 Mathematik und InformatikPhilipps-Universität MarburgMarburgGermany
  2. 2.Dipartimento di Metodi e Modelli Matematici per le Scienze ApplicateUniversità “La Sapienza” in RomaRomaItaly

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