Recent developments in wavelet methods for the solution of PDE's

Silvia Bertoluzza

Bollettino dell'Unione Matematica Italiana (2005)

  • Volume: 8-B, Issue: 3, page 569-590
  • ISSN: 0392-4041

Abstract

top
After reviewing some of the properties of wavelet bases, and in particular the property of characterisation of function spaces via wavelet coefficients, we describe two new approaches to, respectively, stabilisation of numerically unstable PDE's and to non linear (adaptive) solution of PDE's, which are made possible by these properties.

How to cite

top

Bertoluzza, Silvia. "Recent developments in wavelet methods for the solution of PDE's." Bollettino dell'Unione Matematica Italiana 8-B.3 (2005): 569-590. <http://eudml.org/doc/194755>.

@article{Bertoluzza2005,
abstract = {After reviewing some of the properties of wavelet bases, and in particular the property of characterisation of function spaces via wavelet coefficients, we describe two new approaches to, respectively, stabilisation of numerically unstable PDE's and to non linear (adaptive) solution of PDE's, which are made possible by these properties.},
author = {Bertoluzza, Silvia},
journal = {Bollettino dell'Unione Matematica Italiana},
language = {eng},
month = {10},
number = {3},
pages = {569-590},
publisher = {Unione Matematica Italiana},
title = {Recent developments in wavelet methods for the solution of PDE's},
url = {http://eudml.org/doc/194755},
volume = {8-B},
year = {2005},
}

TY - JOUR
AU - Bertoluzza, Silvia
TI - Recent developments in wavelet methods for the solution of PDE's
JO - Bollettino dell'Unione Matematica Italiana
DA - 2005/10//
PB - Unione Matematica Italiana
VL - 8-B
IS - 3
SP - 569
EP - 590
AB - After reviewing some of the properties of wavelet bases, and in particular the property of characterisation of function spaces via wavelet coefficients, we describe two new approaches to, respectively, stabilisation of numerically unstable PDE's and to non linear (adaptive) solution of PDE's, which are made possible by these properties.
LA - eng
UR - http://eudml.org/doc/194755
ER -

References

top
  1. HALL, N. JAWERTH, B. ANDERSSON, L. - PETERS, G., Wavelets on closed subsets of the real line, Technical report, 1993. Zbl0808.42019MR1244602
  2. BAIOCCHI, C. - BREZZI, F., Stabilization of unstable methods, in P.E. Ricci, editor, Problemi attuali dell’Analisi e della Fisica Matematica, Università «La Sapienza», Roma, 1993. MR1249088
  3. BERTOLUZZA, S., Wavelets for the numerical solution of the stokes equation, in Proc. of IMACS ’97 World Conference, Berlin, August 24-29, 1997, 1997, to appear. 
  4. BERTOLUZZA, S., Stabilization by multiscale decomposition, Appl. Math. Lett., 11(6) (1998), 129-134. Zbl0940.65060MR1647121
  5. BERTOLUZZA, S., Analysis of a stabilized three fields domain decomposition method, Technical Report 1175, I.A.N.-C.N.R., 2000, Numer. Math., to appear. Zbl1028.65132MR1961881
  6. BERTOLUZZA, S., Wavelet stabilization of the Lagrange multiplier method, Numer. Math., 86 (2000), 1-28. Zbl0966.65075MR1774008
  7. BERTOLUZZA, S. - CANUTO, C. - TABACCO, A., Stable discretization of convection-diffusion problems via computable negative order inner products, SINUM, 38 (2000), 1034-1055. Zbl0974.65104MR1781214
  8. BERTOLUZZA, S. - KUNOTH, A., Wavelet stabilization and preconditioning for domain decomposition, I.M.A. Jour. Numer. Anal., 20 (2000), 533-559. Zbl0966.65076MR1795297
  9. BERTOLUZZA, S. - VERANI, M., Convergence of a non-linear wavelet algorithm for the solution of PDE’s, Appl. Math. Lett., to appear. Zbl1020.65077MR1938199
  10. BERTOLUZZA, S. - MAZET, S. - VERANI, M., A nonlinear richardson algorithm for the solution of elliptic PDE’s, M3AS, 2003. Zbl1051.65112MR1960998
  11. BRAMBLE, J. H. - LAZAROV, R. D. - PASCIAK, J. E., A least square approach based on a discrete minus one product for first order systems, Technical Report BNL-60624, Brookhaven National Laboratory, 1994. 
  12. BREZZI, F. - FORTIN, M., Mixed and Hybrid Finite Element Methods, Springer, 1991. Zbl0788.73002MR1115205
  13. CANUTO, C. - TABACCO, A. - URBAN, K., The wavelet element method. II. Realization and additional features in 2D and 3D, Appl. Comput. Harmon. Anal., 8, 2000. Zbl0951.42016MR1743533
  14. COHEN, A. - DAHMEN, W. - DE VORE, R., Multiscale decomposition on bounded domains, preprint. Zbl0945.42018
  15. COHEN, A. - DAHMEN, W. - DE VORE, R., Adaptive wavelet methods for elliption operator equations – convergence rates, Technical report, IGPM - RWTH-Aachen, 1998, to appear in Math. Comp. Zbl0980.65130MR1803124
  16. COHEN, A. - DAUBECHIES, I. - VIAL, P., Wavelets on the interval and fast wavelet transforms, ACHA, 1 (1993), 54-81. Zbl0795.42018MR1256527
  17. COHEN, A. - MASSON, R., Wavelet adaptive method for second order elliptic problems: boundary conditions and domain decomposition, Numer. Math., 86(2) (2000), 193-238. Zbl0961.65106MR1777487
  18. DAHLKE, S. - HOCHMUT, R. - URBAN, K., Adaptive wavelet methods for saddle point problems, Technical Report 1126, IAN-CNR, 1999. Zbl0965.65074MR1837765
  19. DAHMEN, W., Stability of multiscale transformations, Journal of Fourier Analysis and Applications, 2(4) (1996), 341-361. Zbl0919.46006MR1395769
  20. DAHMEN, W. - KUNOTH, A. - SCHNEIDER, R., Wavelet least square methods for boundary value problems, Siam J. Numer. Anal., 2002. Zbl1013.65124MR1897946
  21. DAHMEN, W. - MICHELLI, C. A., Using the refinement equation for evaluating integrals of wavelets, SIAM J. Numer. Anal., 30 (1993), 507-537. Zbl0773.65006MR1211402
  22. DAHMEN, W. - SCHNEIDER, R., Composite wavelet bases for operator equations, Math. Comp., 68(228) (1999), 1533-1567. Zbl0932.65148MR1648379
  23. DAUBECHIES, I., Ten lectures on wavelets, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. Zbl0776.42018MR1162107
  24. DEVORE, R. A., Nonlinear approximation, Acta Numerica, 1998. Zbl0931.65007MR1689432
  25. DEVORE, R. A. - JAWERTH, B. - POPOV, V., Compression of wavelet decomposition, Amer. J. Math, 114, 1992. Zbl0764.41024MR1175690
  26. MADAY, Y. - PERRIER, V. - RAVEL, J. C., Adaptivité dynamique sur bases d’ondelettes pour l’approximation d’équations aux derivées partielles, C. R. Acad. Sci Paris, 312 (Série I), (1991), 405-410. Zbl0709.65099MR1096621
  27. MEYER, Y., Wavelets and operators, in Ingrid Daubechies, editor, Different Perspectives on Wavelets, volume 47 of Proceedings of Symposia in Applied Mathematics, pages 35-58, American Math. Soc., Providence, RI, 1993. From an American Math. Soc. short course, Jan. 11-12, 1993, San Antonio, TX. Zbl0810.42015MR1267996
  28. TRIEBEL, H., Interpolation Theory, Function Spaces, Differential Operators, North Holland, 1978. Zbl0387.46032MR503903

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.