Let us first specify the class of problems to be solved with 
-
FEM.
For a more
detailed discussion, see any of the finite element textbooks.
Let 
be an open bounded domain in 
with a
sufficiently
regular boundary 
. The following discussion is
restricted to a class of
problems that can be formulated in the following abstract form:
where X is a subspace of 
, the Sobolev space of first
order, B(.,.)
is a bilinear form on 
and
of the form
In the above formulas, 
are sufficiently
regular functions defined on , and on , respectively.
With a possible extension to multivalued functions numerous examples fall into the category of problems described by the above abstract formulation. To mention a few: linear elliptic problems (both single equations and systems), linear problems resulting from a one-step approximation in time for evolution problems, linear steps of nonlinear problem solution, etc.