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I formulate questions about ion densities in biological and materials problems using Classical Density Functional Theory, as in this paper which is also on the arXiv. The discretization in that paper for the Rosenfeld functional for hard spheres is just pointwise collocation on a regular grid.

I know of no convergence or stability results for this, or any other discretizations of these equations. Do any such results exist? Moreover, has there been a finite element or finite volume formulation of these equations?

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Matt: Have you looked at the polymer self-consistent field theory literature? It's rather similar in spirit to classical DFT's, and discretizations of the system are routinely performed. There's a paper by Alexander-Katz et al. [JCP, 122, 014904 (2005)] that specifically discusses finite-size and discretization effects. That might be a place to start.

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  • $\begingroup$ I looked at this paper. They just do lattice collocation, and spectral methods for the Laplacian, which is exactly what we did in that paper above. They do not prove anything either. $\endgroup$
    – Matt Knepley
    Mar 3, 2012 at 16:31

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