Multiscale adaptive processing for evolution equations

  • A. Cohen
Conference paper


We propose a multiresolution processing strategy which applies to general classes of discretization schemes for evolution equations. The goal of this processing is to reduce significantly the CPU cost and the memory space of the scheme, by making it adaptive, while preserving its order of accuracy. This strategy can be viewed as a combination of the adaptive mesh refinement (AMR) and multiresolution adaptive flux evaluation methods.


Prediction Operator Adaptive Grid Finite Volume Scheme Adaptive Mesh Refinement Haar System 
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  1. [1]
    Abgrall, R. (1997): Multiresolution analysis on unstructured meshes: application to CFD. In: Chetverushkin, B.N. et al. (eds.): Experimentation, modelling and computation in flow, turbulence and combustion. Vol. 2. Wiley, Chichester, pp. 147–156Google Scholar
  2. [2]
    Bécache, E., Joly, P., Tsogka, C. (2000): An analysis of new mixed finite elements for the approximation of wave propagation problems. SIAM J. Numer. Anal. 37, 1053–1084MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    Berger, M., Collela, P. (1989): Local adaptive mesh refinement for shock hydrodynamics. J. Comput. Phys. 82, 64–84MATHCrossRefGoogle Scholar
  4. [4]
    Berger, M., Oliger, J. (1984): Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys. 53, 484–512MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Bertoluzza, S. (1997): An adaptive collocation method based on interpolating wavelets. In: Dahmen, W. et al. (eds.): Multiscale wavelet methods for partial differential equations. Academic Press, San Diego, CA, 109–135CrossRefGoogle Scholar
  6. [6]
    Bihari, B., Harten, A. (1997): Multiresolution schemes for the numerical solution of 2-D conservation laws. I. SIAM J. Sci. Comput. 18, 315–354MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Canuto, C., Cravero, I. (1997): Wavelet-based adaptive finite element method for advection-diffusion equations. Math. Models Methods Appl. Sci. 7, 265–289MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    Cavaretta, A., Dahmen, W., Micchelli, C.A. (1991): Stationary subdivision. Mem. Amer. Math. Soc. 93, no. 453Google Scholar
  9. [9]
    Chiavassa, G., Donat, R. (2001): Point value multiscale algorithms for 2D compressible flows. University of Valencia, Technical report GrAN-99-4Google Scholar
  10. [10]
    Ciarlet, P.G. (1991): Basic error estimates for elliptic problems. In: Ciarlet, P.G., Lions, J.-L. (eds.): Handbook of numerical analysis. Vol. II. North-Holland, Amsterdam, pp. 17–351Google Scholar
  11. [11]
    Cohen, A. (2000): Wavelets in numerical analysis. In: Ciarlet, P.G., Lions, J.-L. (eds.): Handbook of numerical analysis. Vol. VII. North-Holland, Amsterdam, pp. 417–711Google Scholar
  12. [12]
    Cohen, A., Dahmen, W., De Vore, R. (2001): Adaptive wavelet methods for elliptic operator equations: convergence rates. Math. Comp. 70, 27–75MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    Cohen, A., Dahmen, W., De Vore, R. (2002): Adaptive wavelet methods. II. Beyond the elliptic case. Found. Comput. Math. 2, 203–245MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    Cohen, A., Daubechies, I., Feauveau, J.-C. (1992): Biorthogonal bases of compactly supported wavelets. Comm. Pure Appl. Math. 45, 485–560MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    Cohen, A., Dyn, N., Kaber, S.M., Postel, M. (2000): Multiresolution schemes on triangles for scalar conservation laws. J. Comput. Phys. 161, 264–286MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    Cohen, A., Kaber, S.M., Müller, S., Postel, M. (2003): Fully adaptive multiresolution finite volume schemes for conservation laws. Math. Comp. 72, 183–225MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    Cohen, A., Masson, R. (1999): Wavelet adaptive methods for second-order elliptic problems, preconditioning, and adaptivity. SIAM J. Sci. Comput. 21, 1006–1026MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    Dahlke, S., Dahmen, W., Urban, K. (2001): Adaptive wavelet methods for saddle point problems — optimal convergence rates. IGPM Report Nr. 204. RWTH, AachenGoogle Scholar
  19. [19]
    Dahlke, S., DeVore, R. (1997): Besov regularity for elliptic boundary value problems. Comm. Partial Differential Equations 22, 1–16MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    Dahmen, W. (1997): Wavelet and multiscale methods for operator equations. In: Iserles, A. (ed.): Acta Numerica. 1997. Cambridge University Press, Cambridge, pp. 55–228Google Scholar
  21. [21]
    Dahmen, W., Gottschlich-Müller, B., Müller, S. (2001): Multiresolution schemes for conservation laws. Numer. Math. 88, 399–443MathSciNetMATHCrossRefGoogle Scholar
  22. [22]
    Daubechies, I. (1992): Ten lectures on wavelets. SIAM, Philadelphia, PAMATHCrossRefGoogle Scholar
  23. [23]
    De Vore, R. (1998): Nonlinear approximation. In: Iserles, A. (ed.): Acta Numerica. 1998. Cambridge University Press, Cambridge, pp. 51–150Google Scholar
  24. [24]
    De Vore, R., Lucier, B. (1990): High order regularity for conservation laws. Indiana Univ. Math. J. 39, 413–430MathSciNetCrossRefGoogle Scholar
  25. [25]
    Dyn, N. (1992): Subdivision schemes in computer-aided geometric design. In: Light, W.A. (ed.): Advances in numerical analysis. Vol. II. Clarendon Press, OxfordGoogle Scholar
  26. [26]
    Gottschlich-Müller, B., Müller, S. (1999): Adaptive finite volume schemes for conservation laws based on local multi resolution techniques. In: Fey, M., Jeltsch, R. (eds.): Hyperbolic problems: theory, numerics, applications. Vol. I. Birkhäuser, BaselGoogle Scholar
  27. [27]
    Harten, A. (1994): Adaptive multiresolution schemes for shock computations. J. Comput. Phys. 115, 319–338MathSciNetMATHCrossRefGoogle Scholar
  28. [28]
    Harten, A. (1993): Discrete multi-resolution analysis and generalized wavelets. Appl. Numer. Math. 12, 153–192MathSciNetMATHCrossRefGoogle Scholar
  29. [29]
    Harten, A. (1995): Multiresolution algorithms for the numerical solution of hyperbolic conservation laws. Comm. Pure Appl. Math. 48, 1305–1342MathSciNetMATHCrossRefGoogle Scholar
  30. [30]
    Lucier, B. (1986): A moving mesh numerical method for hyperbolic conservation laws. Math. Comp. 46, 59–69MathSciNetMATHCrossRefGoogle Scholar
  31. [31]
    Meyer, Y. (1990): Ondelettes et opérateurs. I. Ondelettes. Hermann, Paris; translated as: Salinger, D.H. (1992): Wavelets and operators. Cambridge University Press, CambridgeMATHGoogle Scholar
  32. [32]
    Sanders, R. (1983): On convergence of monotone finite difference schemes with variable spatial differencing. Math. Comp. 40, 91–106MathSciNetMATHCrossRefGoogle Scholar
  33. [33]
    Sjögreen, B. (1995): Numerical experiments with the multiresolution scheme for the compressible Euler equations. J. Comput. Phys. 117, 251–261MATHCrossRefGoogle Scholar
  34. [34]
    Schröder-Pander, F., Sonar, T. (1995): Preliminary investigations on multiresolution analysis on unstructured grids. DLR Interner Bericht IB 223-95 A 36. DLR, GöttingenGoogle Scholar

Copyright information

© Springer-Verlag Italia 2003

Authors and Affiliations

  • A. Cohen
    • 1
  1. 1.Laboratoire J.-L. LionsUniversité Pierre et Marie CurieParisFrance

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