As we have seen above, in the classical numerical treatment for partial
differential equations - the finite difference method - the solution domain
is approximated by a grid of uniformly spaced nodes. At each node, the
governing differential equation is approximated by an algebraic expression
which references adjacent grid points. A system of equations is obtained
by evaluating the previous algebraic approximations for each node in the
domain. Finally, the system is solved for each value of the dependent
variable at each node. In the Finite Element Method, the solution domain
can be discretized into a number of uniform or non-uniform finite elements
that are connected via nodes. The change of the dependent variable with
regard to location is approximated within each element by an interpolation
function. The interpolation function is defined relative to the values of
the variable at the nodes associated with each element. The original
boundary value problem is then replaced with an equivalent integral
formulation (such as (19)). The interpolation functions are then
substituted into the integral equation, integrated, and combined with the
results from all other elements in the solution domain. The results of
this procedure can be reformulated into a matrix equation of the form
, which is subsequently solved for the unknown variable
[33,25].
