Higher order wavelet-like basis functions in the numerical solution of partial differential equations using the finite element method

Wavelets have become an increasingly popular tool in the computational sciences. They have had numerous applications in a wide range of areas such as signal analysis and data compression and have also been used in the solution of partial differential equations in electromagnetics. Katehi and Krumpholz (1996) have used the Battle-LeMarie wavelets in the multiresolution time domain method and Gordon (1995) has looked previously at wavelet-like functions in the numerical solution of electrostatic problems using the finite element method. In the paper by Gordon, first order wavelet-like functions were generated from the traditional first order basis functions. In this paper, we extend these ideas to third order wavelet-like basis functions. We investigate the effects that these basis functions have on stability and convergence, and compare their performance to that of the first-order wavelet-like functions and the traditional basis functions.