Equation (4.11) is used again.
The three-dimensional finite element method is very similar to
the two-dimensional formulation. Now the analysis is
performed in the three-dimensional domain
with its
boundary
. It leads again to the linear
equation system (4.20). The solution
is approximated by (4.13). The
global matrix
and the right hand side vector
are given analogously to (4.21) by
The expression (4.22) for
is
governed by the Dirichlet boundary condition and
as in (4.17) complies with the
Neumann boundary condition on
. The
global functions
are constructed similarly as in
the two-dimensional case from local ones
defined
only in a few neighbor elements.
As usual, the finite element procedure starts with the
domain discretization. The three-dimensional
volume
is broken into small tetrahedral
elements
. As a consequence the boundary
is subdivided into triangular
elements. This kind of elements are very well suited for
discretizing of arbitrarily or irregularly shaped regions.