Studies on adaptive boundary integral equation methods using wavelets

我们最初计划在Biem中研究在时间域和频率域中使用样条子波来求解波动方程。然而，我们发现，与传统的形状函数相比，这些函数并没有显著提高边界元方程的解的精度和效率。特别是，当考虑问题的因果关系时，发现一些小波函数作为时间形函数不是很方便。因此，我们的结论是，集中于频域方法比时间域方法更合适，并且有必要返回到拉普拉斯方程的更基本的情况。幸运的是，人们发现小波函数的多尺度分析与边界积分解的快速求解方法密切相关，这些方法近年来得到了广泛的研究。小波-Galerkin Biem通过Haar小波函数改进了传统的仅使用尺度函数的Galerkin Biem。由于Haar的小波函数积分为零，采用小波密度函数的单层和双层势能比常规形函数衰减得更快。此外，使用Haar小波函数作为检验函数进一步加速了衰减；实际上，小波-Galerkin方程矩阵中非对角项的衰减率比传统的Galerkin方法大2个数量级。因此，所提出的方法使矩阵方程更加对角占优，并且可以在不降低解质量的情况下用零点代替部分非对角项。我们发现，在约500个自由度的特定问题中，用零替换矩阵中75%的分量是可以接受的。因此，小波-Galerkin Biem是一种非常有前途的BIE的快速求解方法。

We originally planned to investigate the use of spline wavelets in BIEM for wave equations, both in time and frequency domains. However, we found that these functions do not drastically improve accuracy and efficiency of the solutions of BIE compared to the conventional shape functions. In particular some of wavelet functions are found not to be very convenient as time shape functions as one considers the causality of the problem. We thus concluded that it is more appropriate to concentrate on frequency domain approaches than the time domain ones, and that it would be necessary to return to the more fundamental cases of Laplace's equation. Fortunately, it is found that the multiscale analysis of wavelet functions is closely related to fast solution methods of BIE,which are studied extensively these days. We could thus formulate and test the wavelet-Galerkin BIEM,which, in our opinion, is more effective and useful than what we originally intended to investigate.The wavelet-Galerkin BIEM improves the conventional Galerkin BIEM,which uses only scaling functions, by using Haar's wavelet functions. Since Haar's wavelet functions integrate to zero, the single and double layr potentials, with wavelet density functions decay more quickly than with conventional shape functions. In addition the use of Haar's wavelet functions as test functions further accelerates the decay ; indeed, the rate of decay of off-diagonal terms in the matrix of the wavelet-Galerkin equation is bigger by the order of 2 than that of the conventional Galerkin method. Therefore the proposed method makes the matrix equation more diagonally dominated and makes it possible to replace some of off-diagonal terms by zero without deterioration in the quality of the solution. As we found, replacing even 75% of the components in the matrix by zero was acceptable in a certain problem with approximately 500 DOF.We thus conclude that the wavelet-Galerkin BIEM is very promising as a fast solution method of BIE.