Studies on adaptive boundary integral equation methods using wavelets

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.

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.