A second-generation wavelet-based finite element method for the solution of partial differential equations

A new second-generation wavelet (SGW)-based finite element method is proposed for solving partial differential equations (PDEs). An important property of SGWs is that they can be custom designed by selecting appropriate lifting coefficients depending on the application. As a typical problem of SGW algorithm, the calculation of the connection coefficients is described, based on the equivalent filters of SGWs. The formulation of SGW-based finite element equations is derived and a multiscale lifting algorithm for the SGW-based finite element method is developed. Numerical examples demonstrate that the proposed method is an accurate and effective tool for the solution of PDEs, especially ones with singularities.