Let's say we want to add the contributions from two elements which have global numbers for element one and for element two. The local element matrices look like
where the S denotes symmetry. The compenents of represent the connectivities in the global matrix and are the contributions from the ) pair. Thus the two element system would have the following global matrix in terms of the local contributions:
This process continues until we have added in all our element contributions to form the global system. Now we have only to take into account of our boundary conditions and then solve our system.