Mathematical Problems in Engineering
Volume 2009 (2009), Article ID 142384, 18 pages
doi:10.1155/2009/142384
Research Article

A Wavelet Galerkin Finite-Element Method for the Biot Wave Equation in the Fluid-Saturated Porous Medium

Xinming Zhang,1 Jiaqi Liu,2 and Ke'an Liu2

1Harbin Institute of Technology Shenzhen Graduate School, Shenzhen 518055, China
2Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China

Received 19 January 2009; Accepted 26 July 2009

Academic Editor: Victoria Vampa

Copyright © 2009 Xinming Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A wavelet Galerkin finite-element method is proposed by combining the wavelet analysis with traditional finite-element method to analyze wave propagation phenomena in fluid-saturated porous medium. The scaling functions of Daubechies wavelets are considered as the interpolation basis functions to replace the polynomial functions, and then the wavelet element is constructed. In order to overcome the integral difficulty for lacking of the explicit expression for the Daubechies wavelets, a kind of characteristic function is introduced. The recursive expression of calculating the function values of Daubechies wavelets on the fraction nodes is deduced, and the rapid wavelet transform between the wavelet coefficient space and the wave field displacement space is constructed. The results of numerical simulation demonstrate that the method is effective.