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.

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