A modification of the hierarchical basis finite element method is proposed and analyzed by using the wavelet method, see Jaffard [1]. The new basis has local support and results in uniform condition number when applied to discretize partial differential equations. Applications to second order elliptic equations (symmetric or non-symmetric) and Stokes equations (standard or stablized finite elements) are discussed.

[1] S. Jaffard, Wavelets methods for fast resolution of elliptic problems, SIAM J. Numer. Anal., 29 (1992), pp. 965-986.