Error analysis of higher order wavelet-like basis functions in the finite element method

In the computational sciences, both error and wavelet analysis have received abundant attention in the scientific literature. Wavelets have been applied in a wide range of areas such as time-domain analysis, signal compression, and the numerical solution of partial differential equations and integral equations. For instance, wavelet-like basis functions have been used in the numerical solution of differential equations and the error introduced by them has been investigated. Error analysis and Richardson extrapolation have also been used to reduce the numerical error due to the use of first order wavelet-like basis functions. In the present paper, the same techniques will be applied to reduce the numerical error arising when higher order wavelet-like basis functions are used. The numerical error introduced by the higher order wavelet-like basis functions will be discussed. The formation of the Richardson extrapolate, which is found from solutions obtained at different levels of the wavelet analysis, will also be investigated. Finally, a discussion of the error of the Richardson extrapolate will be presented. Example problems will be considered to illustrate these ideas.