$C^1$ spline wavelets on triangulations
HTML articles powered by AMS MathViewer
- by Rong-Qing Jia and Song-Tao Liu PDF
- Math. Comp. 77 (2008), 287-312 Request permission
Abstract:
In this paper we investigate spline wavelets on general triangulations. In particular, we are interested in $C^1$ wavelets generated from piecewise quadratic polynomials. By using the Powell-Sabin elements, we set up a nested family of spaces of $C^1$ quadratic splines, which are suitable for multiresolution analysis of Besov spaces. Consequently, we construct $C^1$ wavelet bases on general triangulations and give explicit expressions for the wavelets on the three-direction mesh. A general theory is developed so as to verify the global stability of these wavelets in Besov spaces. The wavelet bases constructed in this paper will be useful for numerical solutions of partial differential equations.References
- C. de Boor, K. Höllig, and S. Riemenschneider, Box splines, Applied Mathematical Sciences, vol. 98, Springer-Verlag, New York, 1993. MR 1243635, DOI 10.1007/978-1-4757-2244-4
- Zhongying Chen, Charles A. Micchelli, and Yuesheng Xu, A multilevel method for solving operator equations, J. Math. Anal. Appl. 262 (2001), no. 2, 688–699. MR 1859333, DOI 10.1006/jmaa.2001.7599
- Charles K. Chui and Tian Xiao He, Bivariate $C^1$ quadratic finite elements and vertex splines, Math. Comp. 54 (1990), no. 189, 169–187. MR 993926, DOI 10.1090/S0025-5718-1990-0993926-3
- Charles K. Chui and Qingtang Jiang, Surface subdivision schemes generated by refinable bivariate spline function vectors, Appl. Comput. Harmon. Anal. 15 (2003), no. 2, 147–162. MR 2007056, DOI 10.1016/S1063-5203(03)00062-9
- Charles K. Chui and Qingtang Jiang, Refinable bivariate quartic $C^2$-splines for multi-level data representation and surface display, Math. Comp. 74 (2005), no. 251, 1369–1390. MR 2137007, DOI 10.1090/S0025-5718-04-01702-8
- C. K. Chui, J. Stöckler, and J. D. Ward, Compactly supported box-spline wavelets, Approx. Theory Appl. 8 (1992), no. 3, 77–100. MR 1195176
- Charles K. Chui and Jian-zhong Wang, On compactly supported spline wavelets and a duality principle, Trans. Amer. Math. Soc. 330 (1992), no. 2, 903–915. MR 1076613, DOI 10.1090/S0002-9947-1992-1076613-3
- Charles K. Chui and Jian-Zhong Wang, A general framework of compactly supported splines and wavelets, J. Approx. Theory 71 (1992), no. 3, 263–304. MR 1191576, DOI 10.1016/0021-9045(92)90120-D
- Oleg Davydov and Pencho Petrushev, Nonlinear approximation from differentiable piecewise polynomials, SIAM J. Math. Anal. 35 (2003), no. 3, 708–758. MR 2048405, DOI 10.1137/S0036141002409374
- Oleg Davydov and Rob Stevenson, Hierarchical Riesz bases for $H^s(\Omega ),\ 1<s<{5\over 2}$, Constr. Approx. 22 (2005), no. 3, 365–394. MR 2164141, DOI 10.1007/s00365-004-0593-2
- Ronald A. DeVore, Björn Jawerth, and Vasil Popov, Compression of wavelet decompositions, Amer. J. Math. 114 (1992), no. 4, 737–785. MR 1175690, DOI 10.2307/2374796
- Michael S. Floater and Ewald G. Quak, Piecewise linear prewavelets on arbitrary triangulations, Numer. Math. 82 (1999), no. 2, 221–252. MR 1685460, DOI 10.1007/s002110050418
- Michael S. Floater and Ewald G. Quak, Linear independence and stability of piecewise linear prewavelets on arbitrary triangulations, SIAM J. Numer. Anal. 38 (2000), no. 1, 58–79. MR 1770342, DOI 10.1137/S0036142998342628
- T. Goodman and D. Hardin, Refinable multivariate spline functions, in Topics in Multivariate Approximation and Interpolation, K. Jetter et al. (eds.), Elsevier B. V., 2006, pp. 55–83.
- Doug Hardin and Don Hong, Construction of wavelets and prewavelets over triangulations, J. Comput. Appl. Math. 155 (2003), no. 1, 91–109. Approximation theory, wavelets and numerical analysis (Chattanooga, TN, 2001). MR 1992292, DOI 10.1016/S0377-0427(02)00894-4
- Rong Qing Jia, A Bernstein-type inequality associated with wavelet decomposition, Constr. Approx. 9 (1993), no. 2-3, 299–318. MR 1215774, DOI 10.1007/BF01198008
- Rong-Qing Jia, Approximation with scaled shift-invariant spaces by means of quasi-projection operators, J. Approx. Theory 131 (2004), no. 1, 30–46. MR 2103832, DOI 10.1016/j.jat.2004.07.007
- Rong-Qing Jia, Bessel sequences in Sobolev spaces, Appl. Comput. Harmon. Anal. 20 (2006), no. 2, 298–311. MR 2207841, DOI 10.1016/j.acha.2005.11.001
- Rong-Qing Jia and Song-Tao Liu, Wavelet bases of Hermite cubic splines on the interval, Adv. Comput. Math. 25 (2006), no. 1-3, 23–39. MR 2231693, DOI 10.1007/s10444-003-7609-5
- Rong Qing Jia and Charles A. Micchelli, Using the refinement equations for the construction of pre-wavelets. II. Powers of two, Curves and surfaces (Chamonix-Mont-Blanc, 1990) Academic Press, Boston, MA, 1991, pp. 209–246. MR 1123739
- Rong-Qing Jia, Jianzhong Wang, and Ding-Xuan Zhou, Compactly supported wavelet bases for Sobolev spaces, Appl. Comput. Harmon. Anal. 15 (2003), no. 3, 224–241. MR 2010944, DOI 10.1016/j.acha.2003.08.003
- H. Johnen and K. Scherer, On the equivalence of the $K$-functional and moduli of continuity and some applications, Constructive theory of functions of several variables (Proc. Conf., Math. Res. Inst., Oberwolfach, 1976) Lecture Notes in Math., Vol. 571, Springer, Berlin, 1977, pp. 119–140. MR 0487423
- Ming-Jun Lai and Larry L. Schumaker, Macro-elements and stable local bases for splines on Powell-Sabin triangulations, Math. Comp. 72 (2003), no. 241, 335–354. MR 1933824, DOI 10.1090/S0025-5718-01-01379-5
- Song-Tao Liu, Quadratic stable wavelet bases on general meshes, Appl. Comput. Harmon. Anal. 20 (2006), no. 3, 313–325. MR 2224900, DOI 10.1016/j.acha.2005.11.004
- Song-Tao Liu and Yuesheng Xu, Galerkin methods based on Hermite splines for singular perturbation problems, SIAM J. Numer. Anal. 43 (2006), no. 6, 2607–2623. MR 2206450, DOI 10.1137/040607411
- Rudolph Lorentz and Peter Oswald, Criteria for hierarchical bases in Sobolev spaces, Appl. Comput. Harmon. Anal. 8 (2000), no. 1, 32–85. MR 1734847, DOI 10.1006/acha.2000.0275
- P. Oswald, Hierarchical conforming finite element methods for the biharmonic equation, SIAM J. Numer. Anal. 29 (1992), no. 6, 1610–1625. MR 1191139, DOI 10.1137/0729093
- Peter Oswald, Multilevel finite element approximation, Teubner Skripten zur Numerik. [Teubner Scripts on Numerical Mathematics], B. G. Teubner, Stuttgart, 1994. Theory and applications. MR 1312165, DOI 10.1007/978-3-322-91215-2
- M. J. D. Powell and M. A. Sabin, Piecewise quadratic approximations on triangles, ACM Trans. Math. Software 3 (1977), no. 4, 316–325. MR 483304, DOI 10.1145/355759.355761
- Sherman D. Riemenschneider and Zuowei Shen, Wavelets and pre-wavelets in low dimensions, J. Approx. Theory 71 (1992), no. 1, 18–38. MR 1180872, DOI 10.1016/0021-9045(92)90129-C
- Robert C. Sharpley, Cone conditions and the modulus of continuity, Second Edmonton conference on approximation theory (Edmonton, Alta., 1982) CMS Conf. Proc., vol. 3, Amer. Math. Soc., Providence, RI, 1983, pp. 341–351. MR 729339
- Y. Shen and W. Lin, A wavelet-Galerkin method for a hypersingular integral equation system, in Complex Variables: Theory and Applications, R.P. Gilbert (ed.), 2007.
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- Rob Stevenson, A robust hierarchical basis preconditioner on general meshes, Numer. Math. 78 (1997), no. 2, 269–303. MR 1486000, DOI 10.1007/s002110050313
- Rob Stevenson, Stable three-point wavelet bases on general meshes, Numer. Math. 80 (1998), no. 1, 131–158. MR 1642527, DOI 10.1007/s002110050363
- Panayot S. Vassilevski and Junping Wang, Stabilizing the hierarchical basis by approximate wavelets. I. Theory, Numer. Linear Algebra Appl. 4 (1997), no. 2, 103–126. MR 1443598, DOI 10.1002/(SICI)1099-1506(199703/04)4:2<103::AID-NLA101>3.0.CO;2-J
- Jinchao Xu and Wei Chang Shann, Galerkin-wavelet methods for two-point boundary value problems, Numer. Math. 63 (1992), no. 1, 123–144. MR 1182515, DOI 10.1007/BF01385851
- Harry Yserentant, On the multilevel splitting of finite element spaces, Numer. Math. 49 (1986), no. 4, 379–412. MR 853662, DOI 10.1007/BF01389538
Additional Information
- Rong-Qing Jia
- Affiliation: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Canada T6G 2G1
- Email: rjia@ualberta.ca
- Song-Tao Liu
- Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
- Email: songtao@uchicago.edu
- Received by editor(s): July 7, 2004
- Received by editor(s) in revised form: November 29, 2006
- Published electronically: September 12, 2007
- © Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Math. Comp. 77 (2008), 287-312
- MSC (2000): Primary 41A15, 41A63, 42C40, 65D07, 65N30
- DOI: https://doi.org/10.1090/S0025-5718-07-02013-3
- MathSciNet review: 2353954