"It is fair to say that wavelet analysis as it stands has by far not yet reached a steady state in this field. As promising as many of the underlying concepts are, it would be naive to expect their immediate practical sucess in complex, real-life applications governed by PDEs. Current wavelet methods and corresponding software tools are still primarily confined to model problems. While these new ideas may also actually trigger new developments in the context of well-established multilevel techniques, they will ultimately have to compete with the discretization methods and existing software packages for PDEs."
W. Dahmen, A. Kurdila, P. Oswald. Multiscale wavelet methods for Partial Differential Equations. Academic Press, 1997.