The application of Galerkin's method requires the
approximation of the solution h and the weighting function v by
finite series, 
and 
respectively, of the form of equation
(
). This
requires the construction of piecewise defined polynomials (the
basis functions) in simple subdomains 
(the finite element
mesh) of 
.
The domain 
must be divided up so that it can represent
an irregular boundary yet still be simple enough so that
the computation of the solution is manageable. This commonly is achieved
by using triangular or quadrilateral elements.
Consider the generation of the basis
functions by constructing a linear interpolant 
of v of the
form
where the 
's are the basis functions defined in a piecewise
fashion such that they are 1 at exactly one node, decrease linearly to 0
at all adjacent nodes, and are uniformly 0 throughout the rest of the
domain.
In equation (), we see that when this condition
is satisfied, we have
so if 
is chosen to equal 
, then 
exactly
interpolates v at the nodal points. These basis functions can be
constructed by combining pieces of polynomial functions defined locally
over each element. These pieces are called element shape functions.
