The formulation of the finite element approximation starts with the
Galerkin approximation, 
, where 
is our test function.
We now use the finite element method to turn the continuous problems into a
discrete formulation. First we discretize the solution domain,
, and define a finite dimensional subspace,
. One usually defines parameters of the function
at node points,
, i = 0, 1, ..., N. If we now define
the basis functions, 
, as linear continuous piecewise
functions that take the value 1 at node points and zero at other
node points, then we can represent the function 
as:
such that each 
can be written in a unique way
as a linear combination of the basis functions 
. Now
the finite element approximation of the
original boundary value problem can be stated as:
