On the generation and use of two-dimensional wavelet-like basis functions

Wavelets and wavelet analysis have recently become an important area in the computational sciences. Their applications are wide ranging and vary from signal analysis and image compression to the solution of partial differential equations. Gordon and Lee have used wavelet-like basis functions in two-dimensional elliptic problems [1]. In a paper by Jaffard, it was shown that the use of wavelet-like basis functions and a diagonal preconditioner yields a system matrix which has a condition number that is bounded as the number of unknowns is increased [2]. The two-dimensional wavelet-like functions used in this case were derived from the traditional two-dimensional tetrahedral finite element basis functions. Typically, however, higher dimensional wavelets are formed from products of one-dimensional wavelets. So, in this paper, we investigate a new technique for generating the two-dimensional wavelet-like basis functions from products of one-dimensional wavelet-like basis functions. We investigate how this new approach affects the condition number of the system matrix in comparison with the traditional basis set. We also investigate the computational time involved in the generation of the two types of two-dimensional wavelet-like basis functions.