For a problem described by a system of PDEs a local matrix for each element of the discretization , nucleus, is constructed. This nucleus is integrated into the general matrix of the system through a process called assembling. In this section the assembling algorithm applied in numerical schemes of problems discussed in Chapters 3 and 4 is derived.
The Jacobi matrix needed for the Newton method in the case of a finite element discretization is,
The basic nodal function , defined at the arbitrary point is non-zero only on the patch . Furthermore can be represented as,
at the point and at three other points of the tethraedal element .
The discrete operators obtained by testing of -th equation of the system with the basic nodal function is,
Obviously in (2.46) is equal to one of the basic nodal functions at the element . Furthermore, the partial derivative is non-zero only if the stays for one nodal value of .
We now define the operator , , , which assigns a single global index to every local index of vertex belonging to the tethraedal element . The inverse function is also well-defined.
From (2.41), (2.43), and (2.46) we have,
At the end of the assembling process the general matrix contains the values given by (2.40).