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In this paper, a theoretical framework for the least squares finite-element approximation of a fractional order differential equation is presented. Mapping properties for fractional dimensional operators on suitable fractional dimensional spaces are established. Using these properties existence and uniqueness of the least squares approximation is proven. Optimal error estimates are proven for piecewise linear trial elements. Numerical results are included which confirm the theoretical results.

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A three‐dimensional (3‐D) analysis of transport and macrodispersion at the Macrodispersion Experiment (MADE) site [Boggs et al., 1993] using the Fractional Advection‐Dispersion Equation (FADE) developed by Meerschaert et al. [1999, 2001] shows that the Levy dispersion process is scale dependent. Levy dispersion may be superior to Gaussian dispersion on a sufficiently small scale; on larger scales, both theories are likely to suffer from the fact that because of depositional structures most flow fields display an evolving, nonstationary structure. Motion in such fields is advection‐dominated, displays a lot of memory and therefore is not modeled well by Markov random processes which underlie the derivation of both the Gaussian and Levy advection‐dispersion equations [Berkowitz et al., 2002]. To improve plume simulation of an advection‐dominated transport process, one would have to bring in more advective irregularity while simultaneously decreasing the Levy dispersion coefficient. Therefore, on a 3‐D basis, first‐order Levy dispersion has limitations similar to Gaussian dispersion. However, this and related theories, such as the continuous time random walk (CTRW) formalism, are in the early stages of development and thus may be fruitful areas for further research.