After formulating the residue equation including all the boundary contributions, the next task is to evaluate the area and line integrals of Eq. (19.5). For this purpose, the domain is divided into small elements, each of which has a certain number of nodes placed on the boundary and inside, for interpolating field variables.
Figure 19.1: A typical triangular element.
Using a 3-noded triangular element (Fig. 19.1) for our 2D applications, the interpolation of the temperature can be obtained as follows:
where the constants  depend on the nodal coordinates and the nodal temperature value (and therefore vary from element to element). The values of  for a typical element is given by
or, using matrix notation, we obtain
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