Mathematics of Computation

Author(s): Kai Bittner; Karsten Urban.
Journal: Math. Comp. 76 (2007), 903-929.
MSC (2000): Primary 42C40, 35Q30, 41A15
Posted: December 28, 2006
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Abstract: We construct interpolating divergence-free multiwavelets based on cubic Hermite splines. We give characterizations of the relevant function spaces and indicate their use for analyzing experimental data of incompressible flow fields. We also show that the standard interpolatory wavelets, based on the Deslauriers-Dubuc interpolatory scheme or on interpolatory splines, cannot be used to construct compactly supported divergence-free interpolatory wavelets.

1.: C.M. Albukrek, K. Urban, D. Rempfer, and J.L. Lumley, Divergence-free wavelet analysis of turbulent flows, J. Sci. Comput. 17 (2002), no. 1-4, 49-66. MR 1910551
2.: A.Z. Averbuch and V.A. Zheludev, Lifting scheme for biorthogonal multiwavelets originated from Hermite splines, IEEE Trans. Signal Process. 50 (2002), no. 3, 487-500. MR 1895057 (2003b:94004)
3.: G. Battle and P. Federbush, Divergence-free vector wavelets, Michigan Math. J. 40 (1993), no. 1, 181-195. MR 1214063 (94c:42021)
4.: J. Bergh and J. Löfström, Interpolation spaces: An introduction, Springer, Berlin, 1976. MR 0482275 (58:2349)
5.: O. V. Besov, V. P. Il'in, and S. M. Nikol'skii, Integral representation of functions and imbedding theorems, vol. II, Winston & Sons, Washington, D.C., 1979. MR 0521808 (80f:46030b)
6.: Claudio Canuto, Anita Tabacco, and Karsten Urban, The wavelet element method. I. Construction and analysis, Appl. Comput. Harmon. Anal. 6 (1999), no. 1, 1-52. MR 1664902 (99k:42055)
7.: -, The wavelet element method. II. Realization and additional features in 2D and 3D, Appl. Comput. Harmon. Anal. 8 (2000), no. 2, 123-165. MR 1743533 (2001e:42044)
8.: J. M. Carnicer, W. Dahmen, and J. M. Peña, Local decomposition of refinable spaces and wavelets, Appl. Comput. Harmon. Anal. 3 (1996), no. 2, 127-153. MR 1385049 (97f:42050)
9.: A. Cohen, Wavelet methods in numerical analysis, Handbook of numerical analysis, Vol. VII, Handbook Numer. Anal., VII, North-Holland, Amsterdam, 2000, pp. 417-711. MR 1804747 (2002c:65252)
10.: A. Cohen and R. Masson, Wavelet adaptive method for second order elliptic problems: boundary conditions and domain decomposition, Numer. Math. 86 (2000), no. 2, 193-238. MR 1777487 (2001j:65185)
11.: W. Dahmen, B. Han, R.-Q. Jia, and A. Kunoth, Biorthogonal multiwavelets on the interval: Cubic hermite splines, Constr. Approx. 16 (2000), no. 2, 221-259. MR 1735242 (2001a:42036)
12.: Wolfgang Dahmen and Reinhold Schneider, Composite wavelet bases for operator equations, Math. Comp. 68 (1999), no. 228, 1533-1567. MR 1648379 (99m:65122)
13.: -, Wavelets on manifolds. I. Construction and domain decomposition, SIAM J. Math. Anal. 31 (1999), no. 1, 184-230 (electronic). MR 1742299 (2000k:65242)
14.: Wolfgang Dahmen and Rob Stevenson, Element-by-element construction of wavelets satisfying stability and moment conditions, SIAM J. Numer. Anal. 37 (1999), no. 1, 319-352 (electronic). MR 1742747 (2001c:65144)
15.: R. A. DeVore and G. G. Lorentz, Constructive approximation, Springer, New York, 1993. MR 1261635 (95f:41001)
16.: D. Donoho, Interpolating wavelet transforms, Preprint, Standford University, 1992.
17.: P. Federbush, Navier and Stokes meet the wavelet, Comm. Math. Phys. 155 (1993), no. 2, 219-248. MR 1230026 (94g:35171)
18.: S.S. Goh, Q. Jiang, and T. Xia, Construction of biorthogonal multiwavelets using the lifting scheme, Appl. Comput. Harmon. Anal. 9 (2000), no. 3, 336-352. MR 1793422 (2001h:42052)
19.: B. Han, Hermite interpolants and biorthogonal multiwavelets with arbitrary order of vanishing moments, Wavelet Applications in Signal and Image Processing VII (A. Aldroubi, M.A. Unser, and A.F. Laine, eds.), vol. 3813, Proc. SPIE, 1999, pp. 147-161.
20.: B. Han, T.P.-Y. Yu, and B. Piper, Multivariate refinable Hermite interpolant, Math. Comp. 73 (2004), no. 248, 1913-1935 (electronic). MR 2059743 (2005e:41004)
21.: C. Heil, G. Strang, and V. Strela, Approximation of translates of refinable functions, Numer. Math. 73 (1996), 75-94. MR 1379281 (97c:65033)
22.: N.K.-R. Kevlahan and O.V. Vasilyev, An adaptive wavelet collocation method for fluid-structure interaction at high reynolds numbers, SIAM J. Scient. Comput. 26 (2005), 1894-1915. MR 2196581 (2006h:76072)
23.: F. Koster, M. Griebel, N. K.-R. Kevlahan, M. Farge, and K. Schneider, Towards an adaptive wavelet-based 3D Navier-Stokes solver, Numerical flow simulation, I (Marseille, 1997), Notes Numer. Fluid Mech., vol. 66, Vieweg, Braunschweig, 1998, pp. 339-364. MR 1668783 (99i:76123)
24.: J.D. Lakey, P.R. Massopust, and M.C. Pereyra, Divergence-free multiwavelets, Approximation theory IX, Vol. 2 (Nashville, TN, 1998), Innov. Appl. Math., Vanderbilt Univ. Press, Nashville, TN, 1998, pp. 161-168. MR 1744404
25.: J.D. Lakey and M.C. Pereyra, Divergence-free multiwavelets on rectangular domains, Wavelet analysis and multiresolution methods (Urbana-Champaign, IL, 1999), Lecture Notes in Pure and Appl. Math., vol. 212, Dekker, New York, 2000, pp. 203-240. MR 1777994 (2001h:42055)
26.: P.G. Lemarié-Rieusset, Analyses multi-résolutions non orthogonales, commutation entre projecteurs et dérivation et ondelettes vecteurs à divergence nulle, Rev. Mat. Iberoamericana 8 (1992), no. 2, 221-237. MR 1191345 (94d:42044)
27.: -, Un théorème d'inexistence pour les ondelettes vecteurs à divergence nulle, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 8, 811-813. MR 1300948 (95h:42028)
28.: -, Recent developments in the Navier-Stokes problem, Chapman & Hall/CRC Research Notes in Mathematics, vol. 431, Chapman & Hall/CRC, Boca Raton, FL, 2002. MR 1938147 (2004e:35178)
29.: C.A. Micchelli, Interpolatory subdivision schemes and wavelets, J. Approx. Theory 86 (1996), 41-71. MR 1397613 (97g:42030)
30.: L. L. Schumaker, Spline functions: Basic theory, Wiley-Interscience, New York, 1981. MR 606200 (82j:41001)
31.: W. Sweldens, The lifting scheme: A construction of second generation wavelets, SIAM J. Math. Anal. 29 (1998), no. 2, 511-546 (electronic). MR 1616507 (99e:42052)
32.: K. Urban, On divergence-free wavelets, Adv. Comput. Math. 4 (1995), no. 1-2, 51-81. MR 1338895 (96e:42035)
33.: -, Wavelet bases in $H(\rm div)$ and $H(\rm curl)$ , Math. Comp. 70 (2001), no. 234, 739-766. MR 1710628 (2001g:42069)
34.: -, Wavelets in numerical simulation, Lecture Notes in Computational Science and Engineering, vol. 22, Springer-Verlag, Berlin, 2002, Problem adapted construction and applications. MR 1918770 (2003e:42001)
35.: O. V. Vasilyev and N. K.-R. Kevlahan, Hybrid wavelet collocation -- Brinkman penalization method for complex geometry flows, Internat. J. Numer. Methods Fluids 40 (2002), no. 3-4, 531-538, ICFD Conference on Numerical Methods for Fluid Dynamics, Part II (Oxford, 2001). MR 1932995
36.: O.V. Vasilyev and C. Bowman, Second-generation wavelet collocation method for the solution of partial differential equations, J. Comput. Phys. 165 (2000), no. 2, 660-693. MR 1807301 (2002j:65119)
37.: O.V. Vasilyev and N.K.-R. Kevlahan, An adaptive multilevel wavelet collocation method for elliptic problems, J. Comput. Phys. 206 (2005), 412-431. MR 2143325 (2006a:65173)

Retrieve articles in Mathematics of Computation with MSC (2000): 42C40, 35Q30, 41A15

Kai Bittner
Affiliation: University of Ulm, Institute for Numerical Mathematics, Helmholtzstr. 18, D-89069 Ulm, Germany
Email: kai.bittner@uni-ulm.de

Karsten Urban
Affiliation: University of Ulm, Institute for Numerical Mathematics, Helmholtzstr. 18, D-89069 Ulm, Germany
Email: karsten.urban@uni-ulm.de

DOI: 10.1090/S0025-5718-06-01949-1
PII: S 0025-5718(06)01949-1
Keywords: Interpolatory wavelets, divergence-free vector fields
Received by editor(s): March 18, 2005
Received by editor(s) in revised form: March 13, 2006
Posted: December 28, 2006
Additional Notes: We are grateful to Nicolas Kevlahan for bringing our attention to the topic of this paper.
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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