The study of physically realizable problems in two or more dimensions frequently results in partial differential equations which either cannot be solved analytically or lack an exact analytic solution due to the complexity of the boundary conditions or domain. In the case of the partial differential equation governing the flow of ice, the domains in actual applications (e.g. Antarctica, Greenland) are complex and the boundary conditions are also very irregular. While insight has been gained from many analytic analyses, the approximations necessary to obtain a solution often limit the applicability of the solution to generalities or qualitative assessments []. For a realistic, detailed study a numerical method must be used to solve the problem. This model uses the finite element method (FEM).
There are four primary steps that must be undertaken in the use
of the finite element method. They include the following
steps [].
(1) Formulate the problem in a
variational framework in which the
appropriate space 
of admissible
functions is identified.
(2) Construct a finite element mesh
with piecewise-polynomial basis functions
defined on the mesh, which generate a
finite-dimensional subspace of 
.
(3) Construct an approximation of
the variational boundary-value problem on
a finite element subspace 
of
. This
entails the calculation of element
matrices and the generation of a sparse
system of linear algebraic equations in
the values of the approximate solution at
nodal points in the mesh.
(4) Solve the algebraic system.
The above outline for approaching the FEM will be followed in presenting this model. The three subsequent sections discuss in detail the first three of these items. The solution of algebraic systems of equations involves standard numerical matrix methods, and so is not discussed here.