Adaptive Wavelet Galerkin Solution of Some Elastostatics Problems on Irregularly Spaced Nodes

In this paper the second generation wavelets are applied as a basis in finite element method. The wavelet basis is constructed over typical nonequispaced nodes and on boundaries. In addition the wavelet bases are tailored to the Poisson's operator. The wavelet basis is lifted to enforce operator orthogonality, this eliminates coupling between coarse and detail parts of the stiffness matrix. The scale decoupled stiffness matrix permits optimal O(N) computation. The Lagrangian second order wavelets are chosen for demonstration purposes, and a Poisson equation is solved. The potential application of this method in simulating a heterogenous material is outlined.

[1]  W. Sweldens.  The Lifting Scheme: A Custom - Design Construction of Biorthogonal Wavelets "Industrial Mathematics , 1996 .

[2]  S. Mehraeen,et al.  Wavelet Galerkin method in multi‐scale homogenization of heterogeneous media , 2006 .

[3]  John R. Williams,et al.  Wavelet–Galerkin solutions for one‐dimensional partial differential equations , 1994 .

[4]  Harry Yserentant,et al.  Hierarchical bases , 1992 .

[5]  Wolfgang Dahmen,et al.  Local Decomposition of Refinable Spaces and Wavelets , 1996 .

[6]  Wing Kam Liu,et al.  Reproducing kernel particle methods , 1995 .

[7]  Kevin Amaratunga,et al.  Generalized hierarchical bases: a Wavelet‐Ritz‐Galerkin framework for Lagrangian FEM , 2005 .

[8]  Y. Kim,et al.  Adaptive multiscale wavelet-Galerkin analysis for plane elasticity problems and its applications to multiscale topology design optimization , 2003 .

[9]  Zhengjia He,et al.  Adaptive multiresolution finite element method based on second generation wavelets , 2007 .

[10]  I. Daubechies,et al.  Wavelets on irregular point sets , 1999, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[11]  Paolo Venini,et al.  An adaptive wavelet-Galerkin method for an elastic-plastic-damage constitutive model: 1D problem , 2001 .

[12]  T. Belytschko,et al.  Element‐free Galerkin methods , 1994 .

[13]  H. Yserentant.  Erratum. On the Multi-Level Splitting of Finite Element Spaces.(Numer. Math. 49, 379-412 (1986)). , 1986 .

[14]  Eitan Grinspun,et al.  Natural hierarchical refinement for finite element methods , 2003 .

[15]  Harry Yserentant,et al.  On the multi-level splitting of finite element spaces , 1986 .

[16]  Jinchao Xu,et al.  Galerkin-wavelet methods for two-point boundary value problems , 1992 .

[17]  I. Babuska,et al.  The partition of unity finite element method , 1996 .

[18]  G. Strang,et al.  An Analysis of the Finite Element Method , 1974 .