# Numerical solution of Poisson equation with wavelet bases of Hermite cubic splines on the interval

• Jia-wei Xiang (向家伟)
• Xue-feng Chen (陈雪峰)
• Xi-kui Li (李锡夔)
Article

## Abstract

A new wavelet-based finite element method is proposed for solving the Poisson equation. The wavelet bases of Hermite cubic splines on the interval are employed as the multi-scale interpolation basis in the finite element analysis. The lifting scheme of the wavelet-based finite element method is discussed in detail. For the orthogonal characteristics of the wavelet bases with respect to the given inner product, the corresponding multi-scale finite element equation can be decoupled across scales, totally or partially, and suited for nesting approximation. Numerical examples indicate that the proposed method has the higher efficiency and precision in solving the Poisson equation.

## Key words

Poisson equation Hermite cubic spline wavelet lifting scheme wavelet-based finite element method

O351.2

## 2000 Mathematics Subject Classification

65T60 65L05 35F30

## References

1. [1]
Canuto, C., Tabacco, A., and Urban, K. The wavelet element method-part I: construction and analysis. Applied and Computational Harmonic Analysis 6(1), 1–52 (1996)
2. [2]
Canuto, C., Tabacco, A., and Urban, K. The wavelet element method-part II: realization and additional feature in 2D and 3D. Applied and Computational Harmonic Analysis 8(2), 123–165 (2000)
3. [3]
Cohen, A. Numerical Analysis of Wavelet Method, Elsevier, Amsterdam (2003)Google Scholar
4. [4]
Zhou, Y. H., Wang, J. Z., and Zheng, X. J. Application of wavelet Galerkin FEM to bending of beam and plate structures. Appl. Math. Mech. -Engl. Ed. 19(8), 745–755 (1998) DOI: 10.1007/BF02457749
5. [5]
Chen, X. F., He, Z. J., Xiang, J. W., and Li, B. A dynamic multiscale lifting computation method using Daubechies wavelet. Journal of Computational and Applied Mathematics 188(2), 228–245 (2006)
6. [6]
Xiang, J. W., Chen, X. F., He, Z. J., and Dong, H. B. The construction of 1D wavelet finite elements for structural analysis. Computational Mechanics 40(2), 325–339 (2007)
7. [7]
Xiang, J. W., Chen, X. F., He, Z. J., and Zhang, Y. H. A new wavelet-based thin plate element using B-spline wavelet on the interval. Computational Mechanics 41(2), 243–255 (2008)
8. [8]
Xiang, J. W., Chen, X. F., Yang, L. F., and He, Z. J. A class of wavelet-based flat shell elements using B-spline wavelet on the interval and its applications. CMES-Computer Modeling in Engineering and Sciences 23(1), 1–12 (2008)
9. [9]
Mei, S. L., Lu, Q. S., Zhang, S. W., and Jin, L. Adaptive interval wavelet precise integration method for partial differential equations. Appl. Math. Mech. -Engl. Ed. 26(3), 364–371 (2005) DOI: 10.1007/BF02440087
10. [10]
Jin, J. M., Xue, P. X., Xu, Y. X., and Zhu, Y. L. Compactly supported non-tensor product form two-dimension wavelet finite element. Appl. Math. Mech. -Engl. Ed. 27(12), 1673–1686 (2006) DOI: 10.1007/s10483-006-1210-z
11. [11]
He, Y. and Han, B. A wavelet finite-difference method for numerical simulation of wave propagation in fluid-saturated porous media. Appl. Math. Mech. -Engl. Ed. 29(11), 1495–1504 (2008) DOI: 10.1007/s10483-008-1110-y
12. [12]
Basu, P. K., Jorge, A. B., Badri, S., and Lin, J. Higher-order modeling of continua by finite-element, boundary-element, meshless, and wavelet methods. Computers and Mathematics with Applications 46(1), 15–33 (2003)
13. [13]
Jia, R. Q. and Liu, S. T. Wavelet bases of Hermite cubic splines on the interval. Advances in Computational Mathematics 25(1–3), 23–39 (2006)
14. [14]
Quak, E. and Weyrich, N. Decomposition and reconstruction algorithms for spline wavelets on a bounded interval. Applied and Computational Harmonic Analysis 1(2), 217–231 (1994)
15. [15]
Dahmen, W., Kurdila, A., and Oswald, P. Multiscale Wavelet for Partial Differential Equations, Academic Press, San Diego (1997)Google Scholar
16. [16]
Kagan, P., Fischer, A., and Bar-Yoseph, P. Z. Mechanically based models: adaptive refinement for B-spline finite element. International Journal for Numerical Methods in Engineering 57(8), 1145–1175 (2003)

© Shanghai University and Springer Berlin Heidelberg 2009

## Authors and Affiliations

• Jia-wei Xiang (向家伟)
• 1
• 2
Email author
• Xue-feng Chen (陈雪峰)
• 2
• Xi-kui Li (李锡夔)
• 3
1. 1.Faculty of Mechanical and Electrical EngineeringGuilin University of Electronic TechnologyGuilinP. R. China
2. 2.State Key Laboratory for Manufacturing Systems EngineeringXi’an Jiaotong UniversityXi’anP. R. China
3. 3.State Key Laboratory of Structural Analysis for Industrial EquipmentDalian University of TechnologyDalianP. R. China