Creation of sparse boundary element matrices for 2-D and axi-symmetric electrostatics problems using the bi-orthogonal Haar wavelet

In our previous paper we described the creation of sparse boundary element matrices that arise from Laplace's equation with mixed boundary conditions using an orthogonal wavelet basis. In this paper we examine the properties of wavelet expansions, and propose a way to construct even sparser matrices than the ones we obtained using the conventional Haar wavelet. We observe that the number of 'vanishing moments' of a wavelet qualitatively determines the density of BEM matrices. Then we introduce a way of constructing a wavelet that has more vanishing moments than Haar's wavelet, but still retains a form that is very simple to implement. The time to solution employing the new wavelet is comparable to the Haar wavelet; both are faster than algorithms that rely on a conventional piecewise constant basis. However the quality of the solution is much better, for a given sparsity, the L/sub 2/ error of the source distribution is as much as an order of magnitude lower with the new bi-orthogonal wavelet.