A practical determination strategy of optimal threshold parameter for matrix compression in wavelet BEM

摘要

A practical strategy is developed to determine the optimal threshold parameter for wavelet-based boundary element (BE) analysis. The optimal parameter is determined so that the amount of storage (and computational work) is minimized without reducing the accuracy of the BE solution. In the present study, the Beylkin-type truncation scheme is used in the matrix assembly. To avoid unnecessary integration concerning the truncated entries of a coefficient matrix, a priori estimation of the matrix entries is introduced and thus the truncated entries are determined twice: before and after matrix assembly. The optimal threshold parameter is set based on the equilibrium of the truncation and discretization errors. These errors are estimated in the residual sense. For Laplace problems the discretization error is, in particular, indicated with the potential's contribution‖c‖ to the residual norm ‖R‖ used in error estimation for mesh adaptation. Since the normalized residual norm ‖c‖ /‖u‖ (u: the potential components of BE solution) cannot be computed without main BE analysis, the discretization error is estimated by the approximate expression constructed through subsidiary BE calculation with smaller degree of freedom (DOF). The matrix compression using the proposed optimal threshold parameter enables us to generate a sparse matrix with O(N~(1+γ) (0 ≤ γ < 1) non-zero entries. Although the quasi-optimal memory requirements and complexity are not attained, the compression rate of a few per cent can be achieved for N ～ 1000.