The construction method of one-dimensional operator custom-design wavelet finite elements

An important problem of multiscale wavelet algorithm is that the coupling terms of the scaling functions and wavelets increase rapidly as the scale increases, which results in low convergence rate for multiscale wavelet solution. According to the common feature of the operator in engineering structures, a constructive method of operator custom-design wavelet finite elements and its adaptive decoupling algorithm are proposed for engineering structural problems. For the operators defined by the inner product of scaling functions and wavelets, a new kind of one-dimensional operator custom-design wavelets is constructed in multiresolution finite element space based on stable completion. An adaptive operator custom-design wavelet finite element method is presented for engineering structural problems. The advantage of the proposed method is that engineering problems can be solved efficiently by adding operator custom-design wavelets into the local domain while keeping analysis results on the initial scale. Numerical example demonstrates that the decoupling ratio of multiscale stiffness matrix derived by operator custom-design wavelet finite element method is 89.65% and the maximum decreasing ratio of computation complexity for adaptive operator custom-design wavelet finite element method is 49.23%. It is proved that the proposed method is suitable for efficient computation of engineering structural problems.