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,
![]() |
(2.42) |
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).