Studies on adaptive boundary integral equation methods using wavelets

我们最初计划在BIEM中研究样条小波在时域和频域波动方程中的应用。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BIE解的精度和效率。特别是当考虑到问题的因果关系时，发现一些小波函数作为时间形状函数不是很方便。因此，我们得出结论，集中于频域方法比时域方法更合适，并且有必要回到拉普拉斯方程的更基本的情况。所幸的是，人们发现小波函数的多尺度分析与BIE的快速求解方法密切相关，这些方法目前得到了广泛的研究。因此，我们可以制定和测试小波伽辽金BIEM，在我们看来，它比我们最初打算研究的更有效和有用。小波伽辽金BIEM利用Haar小波函数，改进了传统伽辽金BIEM仅使用尺度函数的缺点。由于Haar的小波函数积分为零，单层和双层势，小波密度函数比传统的形状函数衰减更快。此外，使用Haar小波函数作为测试函数进一步加速了衰减；事实上，小波伽辽金方程矩阵中非对角线项的衰减速率比传统伽辽金方法的衰减速率大2个数量级。因此，所提出的方法使矩阵方程更偏向于对角线支配，并且可以在不降低解质量的情况下将一些非对角线项替换为零。正如我们所发现的，在一个大约500自由度的特定问题中，将矩阵中75%的分量替换为零是可以接受的。因此，我们认为小波伽辽金BIEM作为一种快速求解BIE的方法是非常有前途的。

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.