A-posteriori compression of wavelet-BEM matrices

The success of the wavelet boundary element method (BEM) depends on its matrix compression capability. The wavelet Galerkin BEM (WGBEM) based on non-standard form (NS-form) in Tausch (J Numer Math 12(3): 233-254, 2004) has almost linear memory and time complexity. Recently, wavelets with the quasi-vanishing moments (QVMs) have been used to decrease the constant factors involved in the complexity estimates (Xiao in Comput Methods Appl Mech Eng 197:4000-4006, 2008). However, the representations of layer potentials in QVM bases still have much more negligible entries than predicted by a-priori estimates, which are based on the separation of the supports of the source- and test-wavelets. In this paper, we introduce an a-posteriori compression strategy, which is designed to preserve the convergence properties of the underlying Galerkin discretization scheme. We summarize the different compression schemes for the WGBEM and demonstrate their performances on practical problems including Stokes flow, acoustic scattering and capacitance extraction. Numerical results show that memory allocation and CPU time can be reduced several times. Thus the storage for the NS-form is typically less than what is required to store the near-field interactions in the well-known fast multipole method.