The spectral evolution of white dwarfs is governed by diffusion processes which enter into competition with mechanisms such as mass loss, convective mixing, and accretion from the interstellar medium in various phases of the evolution. Until recently, our theoretical understanding of the chemical evolution of these stars has been limited by the very severe numerical difficulties which plague a time-dependent description of the problem. Indeed, diffusion problems in white dwarf interiors and envelopes are particularly demanding from a computational standpoint: they involve relative chemical abundances spanning many orders of magnitude, time integration length of a few billion years, and many physical processes operating with greatly different time constants. We have already introduced in the field a robust numerical technique based on an implicit finite difference scheme designed for nonlinear two-point boundary value problems (Pelletier 1986). This method has been used to investigate a number of problems related to the spectral evolution of white dwarfs (Pelletier 1986; Pelletier et al. 1986; Dupuis et al. 1987). As requirements for further progress in the field become more exacting and in the interest of improving the efficiency, we have sought to develop even more powerful numerical techniques. We briefly introduce here an efficient computational approach to diffusion problems in white dwarfs based on a Galerkin finite element method to solve the convective-diffusion equation in an evolving white dwarf model. As an illustrative example, we discuss some sample results of a detailed investigation of the problem of chemical sedimentation (H, He, and C) in the envelopes of hot white dwarfs and the formation of DA stars.