This paper examines formulation of the discretization schemes for diffusion process simulation that allow coarse grid spacings in the areas of exponentially varying concentrations and fluxes. The method of integral identities is used as a common framework for exponential fitting of both the finite difference and finite element schemes. An exponentially fitted finite difference scheme, with discrete flux terms analogous to those used in Scharfetter-Gummel scheme is justified. An extension of the integral identities in projection form to higher dimensions and a corresponding multidimensional exponentially fitted finite element scheme are proposed. Two-dimensional test computations show clear superiority of the exponentially fitted schemes over the standard approaches as well as robustness of a new finite element scheme regarding irregular grid geometry.