A novel way of using fast wavelet transforms to solve dense linear systems arising from boundary element methods

Abstract

Purpose

The purpose of this paper is to introduce a novel approach to solving linear systems arising from applying a Boundary Element Method (BEM) to elasticity problems.

Design/methodology/approach

The key idea is based on using wavelet transforms as a tool to change dense and fully populated matrices of BEM systems into sparse matrices. Wavelets are then used again to produce an algorithm to solve the resultant sparse linear systems. The wavelet transformation part of the method can be added as a black box to existing BEM codes.

Findings

Numerical results focusing on the sensitivity of the solution for various physical variables to the thresholding parameters, and savings in computer time and memory are presented. The results show that the proposed method is efficient for large problems.

Research limitations/implications

Application of the proposed method is restricted to problems with number of DOF equal to an integer power of 2.

Originality/value

The novel algorithm to solve transformed algebraic linear equations uses NS‐form of the modified matrix, taking the advantage of the hierarchical nature of Multi‐Resolution Analysis (MRA) to decompose a parent system into descendant systems with reduced size. These smaller systems are then solved iteratively using generalized minimal residual method.

Citation

Ebrahimnejad, L. and Attarnejad, R. (2009), "A novel way of using fast wavelet transforms to solve dense linear systems arising from boundary element methods", Engineering Computations, Vol. 26 No. 5, pp. 483-499. https://doi.org/10.1108/02644400910970167