Applying the finite element method to solve a given PDE leads to an algebraic system of equations. In order to solve this system of equations, the global stiffness matrix,
, and the load vector,
, have to be determined. However, instead of computing them using directly (4.11) and (4.12), in practice they are computed by summing the contributions from the different elements [152,153,154] according to
Note that
unless both
and
belong to the same element
. Thus, the calculations (4.13) and (4.14) can be limited to the nodes of the element
, so that
, where
is the number of vertices of the element. In this way, for each element
, a
matrix is obtained, which is called element stiffness or nucleus matrix. Thus, the general system matrix,
, can be computed by first computing the nucleus matrices for each
and then summing the contributions from each element according to (4.13) [152]. The right-hand side vector,
, is computed in the same way.
This process of constructing the general system matrix is called assembly [152].
The main advantage of this assembly process is that it greatly simplifies the computation of the system matrix and right-hand side vector, since (4.11) and (4.12) can be easily calculated for each element of the domain discretization.