Equations such as (
) and () are called the
``strong'' or classical formulation of the problem. These
differential equations must be satisfied at every point within the
domain. Solutions for this type of formulation are often impossible
because of irregularities and discontinuities in the data describing the
problem. For instance, there may be abrupt changes in the various
coefficients of equations () and () that do not
allow for simple solutions. Hence we must look for a ``weaker''
variational formulation of the problem that will allow for such
irregularities in the data.
Consider the steady-state version of the 1-dimensional equation
(), and form the residual error function, r, given by the
following expression.
We can multiply this by an arbitrary trial function, v, and integrate
over the whole domain, 
. This can be done only over
regions where the data of the problem is smooth. Where the data is
insufficiently smooth, we can break the integral into parts spanning
smooth regions on either side of the discontinuity. The smooth
subdomains are the so-called ``elements'' of the finite-element
method. An integral over one of the subdomains, 
, will be
equal to zero, and can be integrated by parts so that the
term becomes symmetric in u and v.
If we have a discontinuity at 
, a point between 0 and L, we
will have from equation ()
where 
is the ``smooth part'' of the source function, f, and
the jump condition expressing the discontinuity in 
at 
is
defined by the following.
Equation () expresses the ease with which general boundary
conditions can be specified in the finite-element method. For the case of
essential boundary conditions (
and 
), we simply
require that the trial function, v, be precisely zero at the
boundaries, and we impose the specified boundary values in the matrix
formulation. For natural boundary conditions where the flux is
specified at the boundary (
or
) we simply include these boundary terms
explicitly as the second and fourth terms of the right-hand side of
equation (). More complex boundary conditions involving
linear combinations of the boundary flux and an ambient state variable
value can be specified in an analogous manner.
