A domain decomposition method for biharmonic equation

Abstract

The domain decomposition method proposed by Schwarz [1] is gaining significance in view of the possibilities for parallel implementation. In this paper, we apply the domain decomposition method to the biharmonic equation in two overlapping discs. In each subdomain, a boundary integral formula is used to obtain the solution. Unlike the finite element or finite difference methods where the iterations have to be performed on the entire domain, the iterations of the boundary integral formulation need to be done only at the discretized points of the boundaries. Another aspect of the paper is that, in order to handle the operations with dense matrices arising from the boundary integral formulation, the discrete wavelet transform is used to compress the matrices which results in reduced computations without loss of accuracy. Examples involving a Laplace equation in an L-shaped region and a domain bounded by a cardioid and a circle are also illustrated.