Here we're going to cover (a) where the ice-shelf equations come from, and (b) provide a brief overview of one method (the finite element method) which can be profitably applied to solve these equations.
These slides pdf illustrate how ice shelves are different from grounded ice sheets. They also make a point that as everything on the Earth, ice sheets and ice shelves are much longer and wider than thicker. Derivations of the zero- and first-order stress-balance equations for ice shelves is described here pdf. It is strongly recommended to repeat all derivations at least one more time. The whole derivation followed a paper by Doug MacAyeal (1989) pdf
FEM notes -- Poisson's equation on the Rosspdf
A sample matlab script is here sample_poisson_v1.m.gz
A FEM mesh for the Ross Ice Shelf is here mesh_RIS.mat.gz