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?