The first step in implementing the FEM is the formulation
of the problem in a weak or variational form. To arrive at the weak
formulation of the
problem, equation (
) must be modified so that it is satisfied
in an average sense over the domain 
. Integrating equation
()
with an arbitrary weighting function 
yields the following equation
which holds for all functions 
such that the values of the integrals
exist.
It is advantageous to cast the problem in a symmetric form so that the solution, h, can belong to the same class of functions as the weighting functions, v. In addition, natural boundary conditions are easier to implement in the symmetric formulation.
From the product rule for the gradient, we have
which when applied to equation () results in
The last integral on the right hand side can be converted to a
boundary integral using Gauss's divergence formula.
The boundary of the domain may
be divided into segments dependent upon the type of boundary
condition applied. The notation 
is used
for portions of the boundary where essential boundary conditions
are applied (state variable h is specified). Along 
the trial function v is taken to be zero. For portions of the boundary
where natural boundary conditions are applied (flux specified),
the notation 
is used.
Applying the divergence theorem to the weak
statement 
(), we have
where (
) is the normal component of the
ice flux density across a portion of the boundary of the domain,
. It is clear that natural boundary conditions
are immediately incorporated
into the solution in the last term of equation(). The
variational statement is now symmetric with respect to h and v.
Hence all differentiability and integrability requirements on h and
v are the same, allowing use of the same class of functions for both
h and v.
By definition the space of admissible functions,
, is a linear space
of functions, is infinite-dimensional, and
has a countably infinite basis. Therefore, there exists a
countable infinite set of functions (
, 
,
,...) in 
, such that any v that belongs to
can be written in the form
where the 
's are constants. The set 
is called a
basis and the individual 
's are called basis functions.
Consider an approximation 
of v by taking
a finite number of terms in the series of equation (). That
is,
Consideration of the approximation using N basis functions
results in the simplification of the class of admissible
functions to an N-dimensional subspace 
of
. The variational
boundary-value problem is stated in the following form:
The problem is now in an acceptable form for the construction of approximate solutions using Galerkin's Method and the FEM.
