Resistor networks

Main reference is Electric circuit diagram equivalents of fields by Gerard Westendorp. Also of interest may be the Lumped-element model.
Crucial is the following. According to Kirchhoff's circuit laws a basic property of resistor networks is that the net current to a nodal point / entering a junction $(i)$ is zero: $$ \sum_j I_{ij} = 0 \quad \Longleftrightarrow \quad \sum_j (V_j-V_i)/R_{ij} = 0 $$ Working out for separate resistors $R$ , it can be seen that the equations are a finite Element assembly of symmetric $2\times 2$ matrices: $$ {\bf E} = \begin{bmatrix} +1/R & -1/R \\ -1/R & +1/R \end{bmatrix} $$ Thus we have landed on solid ground, according to the Definition of finite elements (Ciarlet).
Software has been developed for the purpose:

Examples

inlezen netS1 netS2


Bulk: 
1 2 3.
2 3 10.
3 4 5.
Prescribed: 
4 1 9.
Output: 
# unknowns = 4
bandwidth = 2
Resistors:
    1    2   3.00000
    2    3  10.00000
    3    4   5.00000
Prescribed:
    4    1   9.00000
Voltages:
    1   9.00000
    2   7.50000
    3   2.50000
    4   0.00000
Amperages:
    1    2   0.50000
    2    3   0.50000
    3    4   0.50000
    1    4   0.50000
Resistance:
    1    4  18.00000
Dissipation =    4.50000

inlezen netP1 netP2


Bulk: 
1 2 10.
1 2 2.
1 2 1.
Prescribed: 
2 1 9.
Output: 
# unknowns = 2
bandwidth = 2
Resistors:
    1    2  10.00000
    1    2   2.00000
    1    2   1.00000
Prescribed:
    2    1   9.00000
Voltages:
    1   9.00000
    2   0.00000
Amperages:
    1    2   0.90000
    1    2   4.50000
    1    2   9.00000
    1    2  14.40000
Resistance:
    1    2   0.62500
Dissipation =  129.60000

inlezen netP1a netP2a


Bulk: 
2 3 10.
2 3 2.
2 3 1.
1 2 0.
Prescribed: 
3 1 9.
Output: 
# unknowns = 3
bandwidth = 2
Resistors:
    2    3  10.00000
    2    3   2.00000
    2    3   1.00000
    1    2   0.00000
Prescribed:
    3    1   9.00000
Voltages:
    1   9.00000
    2   9.00000
    3   0.00000
Amperages:
    2    3   0.90000
    2    3   4.50000
    2    3   9.00000
    1    2  14.40000
    1    3  14.40000
Resistance:
    1    3   0.62500
Dissipation =  129.60000