Comparison between wavelet and fast multipole data sparse approximations for Poisson and kinematics boundary - domain integral equations

The boundary element method applied on non-homogenous partial differential equations requires calculation of a fully populated matrix of domain integrals. This paper compares two techniques: the fast multipole method and the fast wavelet transform, which are used to reduce the complexity of such domain matrices. The employed fast multipole method utilizes the expansion of integral kernels into series of spherical harmonics. The wavelet transform for vectors of arbitrary length, based on Haar wavelets and variable thresholding limit, is used. Both methods are tested and compared by solving the scalar Poisson equation and the velocity-vorticity vector kinematics equation. The results show comparable accuracy for both methods for a given data storage size. Wavelets are somewhat better for high and low compression ratios, and the fast multipole methods gives better results for moderate compressions. Considering implementation of the methods, the wavelet transform can easily be adapted for any problem, while the fast multipole method requires different expansion for each integral kernel.