Tree Algorithms in Wavelet Approximations by Helmholtz Potential Operators

Abstract

By means of the limit and jump relations of classical potential theory with respect to the Helmholtz equation a wavelet approach is established on a regular surface. The multiscale procedure is constructed in such a way that the emerging potential kernels act as scaling functions, wavelets are defined via a canonical refinement equation. A tree algorithm for fast computation of a function discretely given on a regular surface is developed based on numerical integration rules. By virtue of the tree algorithm, an efficient numerical method for the solution of Fredholm integral equations involving boundary-value problems of the Helmholtz equation corresponding to (general) regular (boundary) surfaces is discussed in more detail.

Keywords:

• Regular surface
• Helmholtz layer potential operators
• Wavelets
• Tree algorithms
• Integral equations

Acknowledgment

The Geomathematics Group would like to thank the Graduiertenkolleg “Mathematik und Praxis,” University of Kaiserslautern, and the “Stiftung Rheinland-Pfalz für Innovation,” Mainz, for financial support.