 # Finite Element Method (FEM) Local and global graph numbering, taken from [*1].

## Vertex connections -> Global System of equations

Two factors have to be accounted for when elements are connected to form the global system of equations:

(1)    Continuity of the primary variables (here: temperature) across element bounds.

(2)    Mass balance of secondary variables (here: heat conduction) for the element.

The principles of connecting elements shall be shown with the help of an example taken from figure to the right.

The following relations between local and global values for the temperature T can be derived from the figure:

Eq. 3-30 represents the first of the above requirements for the connecting of elements. If the vertex values of the adjacent vertices of elements 1 and 2 are identical, the continuity requirement at their common edge is met ( of course only if the approximation functions of element 1 and 2 are of the same order).

Now we look at the mass balance of the two elements. The following relationships hold for the heat conduction qne e.g. from element 1 to element 2 (and vice versa):

The heat conduction qne  in Eq. 3-31 has to be positive pointing away from the edge of the element if we cycle counter-clockwise through the elements, according to the sign-convention.  