A novel approach to solve linear systems arising from BEM using fast wavelet transforms

Abstract

A novel approach to solve linear systems arising from BEM using fast wavelet transforms Latif Ebrahimnejad * † and Reza Attarnejad Communicated by V. Volpert This paper uses Daubechies orthogonal wavelets to change dense and fully populated matrices of boundary element method (BEM) systems into sparse and semi-banded matrices. Then a novel algorithm based on hierarchical nature of multiresolution analysis is introduced to solving resultant sparse linear systems. This algorithm decomposes NS-form of transformed parent matrix into descendant systems with reduced sizes and solves them iteratively using GMRES algorithm. Both parts, changing dense matrices to sparse systems and the novel solver, can be added as a black box to the existing BEM codes. Transforming matrices into wavelet space needs less time than saved by solving sparse large systems. Numerical results with a precise study on sensitivity of solution for physical variables to the thresholding parameter, and savings in computer time and memory are presented. Also, the suitable value for thresholding parameter is recommended for elasticity problems. The results indicate that the proposed method is efficient for large problems. Copyright © 2009 John Wiley & Sons, Ltd.Keywords: boundary element method; wavelet transform; MRA; GMRES solver; wavelet solver IntroductionBoundary element method (BEM) can be considered as the eldest mesh reduction method. BEM is an efficient boundary type method, where only the boundary of domain is required to be discretized. Then the dimension of problem is reduced by one in comparison with FE and FD [1,2]. Although total degrees of freedom for BE is much less than FE and FD, because of arising dense and fully populated matrices instead of sparse and banded matrices in FE and FD, the efficiency of BEM for large problems is decreased considerably.Wavelets were first used by Beylkin, Coifman and Rokhlin (BCR) [3] to change the dense and fully populated matrices of BEM. Then wavelet compression was applied to BEM for Laplace equation by . In the earlier works, authors have developed the method to elasticity problems [7, 8]. Permutation technique introduced by Chen [9] changes finger-like matrices of transformed space to semi-banded matrices.Applying wavelet compression to BEM can be implemented as a black box in existing BEM codes. While standard BEM uses Gaussian elimination with O(N 3 ) operations to solve algebraic linear equations, where N is total number of equations, iterative solvers such as Lanczos, CG and GMRES need O(N 2 ) operations per iteration [10--13].Wavelet solver can reduce complexity order to O((N Log N) ), which is impressively motivating. In this paper Daubechies orthogonal wavelets are used to transform original system matrix to a semi-banded sparse one, which then is solved by using wavelet solver. This solver solves linear algebraic systems using NS-form of transformed matrix, hierarchical nature of multiresolution analysis (MRA) and GMRES algorithm.In what follows, required equations used in BEM for elastic...

Development and implementation of some BEM variants—A critical review

“…There are dozens wavelets that can be chosen and primarily categorized as either orthogonal [63,64] or biorthogonal [87,88]. The selection of wavelets depends on their characteristics, mainly on orthogonality, vanishing moments and smoothness.…”