Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Wavelet bases in $\mathbf{H}( \mathrm{div})$ and $\mathbf{H}(\mathbf{curl})$


Author: Karsten Urban
Journal: Math. Comp. 70 (2001), 739-766
MSC (2000): Primary 65T60; Secondary 35Q60, 35Q30
DOI: https://doi.org/10.1090/S0025-5718-00-01245-X
Published electronically: May 19, 2000
MathSciNet review: 1710628
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract:

Some years ago, compactly supported divergence-free wavelets were constructed which also gave rise to a stable (biorthogonal) wavelet splitting of $\mathbf{H}(\mathrm{div};\Omega)$. These bases have successfully been used both in the analysis and numerical treatment of the Stokes and Navier-Stokes equations. In this paper, we construct stable wavelet bases for the stream function spaces $\mathbf{H}(\mathbf{curl};\Omega)$. Moreover, $\mathbf{curl}$-free vector wavelets are constructed and analysed. The relationship between $ \mathbf{H}(\mathrm{div};\Omega)$ and $\mathbf{H}(\mathbf{curl};\Omega)$ are expressed in terms of these wavelets. We obtain discrete (orthogonal) Hodge decompositions.

Our construction works independently of the space dimension, but in terms of general assumptions on the underlying wavelet systems in $L^2(\Omega)$ that are used as building blocks. We give concrete examples of such bases for tensor product and certain more general domains $\Omega\subset\mathbb{R}^n$. As an application, we obtain wavelet multilevel preconditioners in $\mathbf{H}(\mathrm{div};\Omega)$ and $\mathbf{H}(\mathbf{curl};\Omega)$.


References [Enhancements On Off] (What's this?)

  • 1. D.N. Arnold, R.S. Falk, and R. Winther, Preconditioning in $H(\operatorname{div})$ and applications, Math. Comput. 66, No. 219 (1997), 957-984. MR 97i:65177
  • 2. D.N. Arnold, R.S. Falk, and R. Winther, Multigrid in $H(\operatorname{div})$ and H $(\mathbf{curl})$, Dept. of Math., Penn State Univ., Preprint, 1998.
  • 3. A. Barinka, T. Barsch, K. Urban, and J. Vorloeper, The Multilevel Library: Software Tools for Multiscale Methods and Wavelets, Version 1.0, Documentation, RWTH Aachen, IGPM Preprint 156, 1998.
  • 4. R. Beck, P. Deuflhard, R. Hiptmair, R.H.W. Hoppe, and B. Wohlmuth, Adaptive multilevel methods for edge element discretizations of Maxwell's equations, ZIB Berlin, Report SC-97-66, 1997, to appear in Surveys of Mathematics in Industry.
  • 5. G. Beylkin, R. Coifman, and V. Rokhlin, Fast wavelet transforms and numerical algorithms, Comm. Pure Appl. Math. 44 (1991), 141-183. MR 92c:65061
  • 6. A. Bossavit, Computational Electromagnetism, Academic Press, San Diego, 1998. MR 99m:78001
  • 7. F. Brezzi and L.D. Marini, A three-field domain decomposition method, Contemp. Math. 157 (1994), 27-34. MR 95a:65202
  • 8. C. Canuto, M.Y. Hussaini, A. Quarteroni, and T.A. Zang, Spectral Methods in Fluid Dynamics, Springer-Verlag, New York, 1988. MR 89m:76004
  • 9. C. Canuto, A. Tabacco, and K. Urban, The wavelet element method, part I: Construction and analysis, Appl. Comp. Harm. Anal. 6 (1999), 1-52. MR 99k:42055
  • 10. C. Canuto, A. Tabacco, and K. Urban, The wavelet element method, part II: Realization and additional features in 2d and 3d, Preprint 1052, Instituto di Analisi Numerica del C.N.R., Pavia, 1997. To appear in Appl. Comp. Harm. Anal.
  • 11. A. Cohen, Wavelet methods in Numerical Analysis, in: Handbook of Numerical Analysis, North Holland, Amsterdam, to appear.
  • 12. A. Cohen, I. Daubechies, and J.-C. Feauveau, Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math. 45 (1992), 485-560. MR 93e:42044
  • 13. A. Cohen, I. Daubechies, and P. Vial, Wavelets on the interval and fast wavelet transforms, Appl. Comp. Harm. Anal. 1 (1993), 54-81. MR 94m:42074
  • 14. A. Cohen and R. Masson, Wavelet methods for second order elliptic problems -- preconditioning and adaptivity, Preprint, Univ. P. et M. Curie, Paris, 1997.
  • 15. A. Cohen and R. Masson, Wavelet adaptive method for second order elliptic problems -- boundary conditions and domain decomposition, Preprint, Univ. P. et M. Curie, Paris, 1997.
  • 16. W. Dahmen, Wavelet and multiscale methods for operator equations, Acta Numerica 6 (1997), 55-228. MR 98m:65102
  • 17. W. Dahmen, A. Kunoth, and K. Urban, A wavelet Galerkin method for the Stokes problem, Computing 56 (1996), 259-301. MR 97g:65228
  • 18. W. Dahmen, A. Kunoth, and K. Urban, Biorthogonal spline wavelets on the interval--stability and moment conditions, Appl. Comp. Harm. Anal. 6 (1999), 132-196. MR 99m:42046
  • 19. W. Dahmen and R. Schneider, Composite wavelet bases for operator equations, Math. Comput. 68 (1999), 1533-1567. MR 99m:65122
  • 20. W. Dahmen and R. Schneider, Wavelets on Manifolds I: Construction and Domain Decomposition, RWTH Aachen, IGPM Preprint 149, 1998. To appear in SIAM J. Math. Anal.
  • 21. W. Dahmen and R. Schneider, Wavelets with complementary boundary conditions -- Functions spaces on the cube, Results Math. 34 (1998), 255-293. MR 99h:42057
  • 22. I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics 61, 1992. MR 93e:42045
  • 23. C.A.J. Fletcher, Computational Techniques for Fluid Dynamics, Volume II, Springer-Verlag, Berlin, 1988. MR 90c:76006b
  • 24. R. Hiptmair, Multilevel Preconditioning for Mixed Problems in Three Dimensions, PhD-Thesis, Univ. of Augsburg, 1996.
  • 25. V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes-Equations, Springer-Verlag, Berlin, 2nd edition, 1986. MR 88b:65129
  • 26. P.G. Lemarié-Rieusset, Analyses multi-résolutions non orthogonales, commutation entre projecteurs et derivation et ondelettes vecteurs à divergence nulle (in French), Rev. Mat. Iberoamericana 8 (1992), 221-237. MR 94d:42044
  • 27. P.G. Lemarié-Rieusset, Un théorème d'inexistence pour les ondelettes vecteurs à divergence nulle (in french), C.R. Acad. Sci. Paris Sér. I Math. 319 (1994), 811-813. MR 95h:42028
  • 28. R. Masson, Wavelet discretizations of the Stokes problem in velocity-pressure variables, Preprint, Univ. P. et M. Curie, Paris, 1998.
  • 29. Y. Meyer, Ondelettes et Opèrateurs, vol. I (in french), Hermann, Paris, 1990.
  • 30. J.E. Roberts and J.-M. Thomas, Mixed and hybrid methods, in: Handbook of Numerical Analysis, Vol. II, P.G. Ciarlet and J.L. Lions (eds.), Elsevier Science Publishers, North-Holland, 1991, 523-640. CMP 91:14
  • 31. G. Strang and T. Nguyen, Wavelets and Filter Banks, Wellesley Cambridge Press, Cambridge, 1996. MR 98b:94003
  • 32. K. Urban, On divergence-free wavelets, Adv. Comput. Math. 4 (1995), 51-82. MR 96e:42035
  • 33. K. Urban, Multiskalenverfahren für das Stokes-Problem und angepaßte Wavelet-Basen (in german), PhD thesis, Verlag der Augustinus-Buchhandlung, Aachen, 1995.
  • 34. M.V. Wickerhauser, Adapted Wavelet Analysis from Theory to Software, AK Peters, Wellesley, 1994. MR 95j:94005

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65T60, 35Q60, 35Q30

Retrieve articles in all journals with MSC (2000): 65T60, 35Q60, 35Q30


Additional Information

Karsten Urban
Affiliation: RWTH Aachen, Institut für Geometrie und Praktische Mathematik, Templergraben 55, 52056 Aachen, Germany
Address at time of publication: Istituto di Analisi Numerica del C.N.R., via Abbiategrasso 209, 27100 Pavia, Italy
Email: urban@igpm.rwth-aachen.de

DOI: https://doi.org/10.1090/S0025-5718-00-01245-X
Keywords: $\mathbf{H}(\mathrm{div};\Omega)$, $\mathbf{H}(\mathbf{curl};\Omega)$, stream function spaces, wavelets
Received by editor(s): January 4, 1999
Received by editor(s) in revised form: May 24, 1999
Published electronically: May 19, 2000
Additional Notes: I am very grateful to Franco Brezzi and Claudio Canuto for fruitful and interesting discussions as well as helpful remarks. This paper was written when the author was in residence at the Istituto di Analisi Numerica del C.N.R. in Pavia, Italy.
This work was supported by the European Commission within the TMR project (Training and Mobilty for Researchers) Wavelets and Multiscale Methods in Numerical Analysis and Simulation, No. ERB FMRX CT98 0184 and by the German Academic Exchange Service (DAAD) within the Vigoni–Project Multilevel–Zerlegungsverfahren für Partielle Differentialgleichungen.
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society