Two-dimensional higher-order wavelet-like basis functions in the finite element method

Wavelets and wavelet analysis have recently become increasingly important in the computational sciences. Wavelets have many applications in areas such as signal analysis, image compression, and the numerical solution of partial differential equations and integral equations. For instance, two-dimensional wavelet-like basis functions are used in the numerical solution of elliptic partial differential equations. With diagonal preconditioning, the use of these wavelet-like basis functions yields a system matrix that has a condition number that is bounded by a constant as the number of basis functions is increased. These two-dimensional wavelet-like functions are derived directly from the traditional two-dimensional first order finite element basis functions. Typically, however, higher dimensional wavelets are formed from products of one-dimensional wavelets. Thus, in this paper we consider this alternative method for generating wavelet-like basis functions. Specifically, we investigate the generation of higher order two-dimensional wavelet-like basis functions from products of higher order one-dimensional wavelet-like basis functions. A discussion of the advantages and disadvantages of this new approach will be presented.