I'm working through an FEM calculation by hand to verify I understand the algorithm before I write a bunch of code based on incorrect assumptions. However, when I iterate through the calculations, the values diverge. Can I diagnose my issue by inspecting the coefficient matrix or problem setup for concordance with some known properties?
Setup
Geometry
The shape is a trapezoid divided into three triangular elements, for which I have approximated the value (potential) within with a linear function:
For the sake of an example problem, I assigned the following coordinates and values to the global indices:
| Node | Coordinate | Value (V_i) |
|------+------------+-------------|
| 1 | 1,1 | 100 |
| 2 | 0,0 | 100 |
| 3 | 2,1 | * |
| 4 | 1,0 | * |
| 5 | 3,0 | * |
I computed the area for each triangle $i$ as
$$ 2A_i = (x^{(i)}_1 y^{(i)}_2 - x^{(i)}_2 y^{(i)}_1) + (x^{(i)}_3 y^{(i)}_1 - x^{(i)}_1 y^{(i)}_3) + (x^{(i)}_2 y^{(i)}_3 - x^{(i)}_3 y^{(i)}_2) $$
where the indices are the local index labels inside each triangle.
I notice that if I place the origin in the upper left, the counter-clockwise labeling no longer produces positive $A$ for triangles. However, as I understand it, that's only a convention and shouldn't affect the calculation.
| Triangle | 2A |
|----------|------|
| 1 | -1 |
| 2 | -1 |
| 3 | -2 |
Coefficient Matrix
The coefficient matrix for this problem is given by the symmetric form
\begin{equation} C = \begin{bmatrix} C_{11}^{(1)} + C_{11}^{(2)} && C_{13}^{(1)} && C_{12}^{(2)} && C_{12}^{(1)} + C_{13}^{(2)} && 0 \\ && C_{33}^{(1)} && 0 && C_{23}^{(1)} && 0 \\ && && C_{22}^{(2)} + C_{11}^{(3)} && C_{23}^{(2)} + C_{13}^{(3)} && C_{12}^{(3)} \\ && && && C_{22}^{(1)} + C_{33}^{(2)} + C_{33}^{(3)} && C_{32}^{(3)} \\ && && && && C_{22}^{(3)} \end{bmatrix} \end{equation}
where the local terms are
\begin{align} C_{11}^{(e)} &= \frac{1}{4A} \left[ (y_2 - y_3)^2 + (x_3 - x_2)^2 \right] \\ C_{12}^{(e)} &= \frac{1}{4A} \left[ (y_2 - y_3)(y_3 - y_1) + (x_3 - x_2)(x_1 - x_3) \right] \\ C_{13}^{(e)} &= \frac{1}{4A} \left[ (y_2 - y_3)(y_1 - y_2) + (x_3 - x_2)(x_2 - x_1) \right] \\ C_{21}^{(e)} &= C_{12}^{(e)} \\ C_{22}^{(e)} &= \frac{1}{4A} \left[ (y_3 - y_1)^2 + (x_1 - x_3)^2 \right] \\ C_{23}^{(e)} &= \frac{1}{4A} \left[ (y_3 - y_1)(y_1 - y_2) + (x_1 - x_3)(x_2 - x_1) \right] \\ C_{31}^{(e)} &= C_{13}^{(e)} \\ C_{32}^{(e)} &= C_{23}^{(e)} \\ C_{33}^{(e)} &= \frac{1}{4A} \left[ (y_1 - y_2)^2 + (x_2 - x_1)^2 \right] \end{align}
which I have computed to be (symmetric terms not shown):
\begin{equation} \begin{bmatrix} -0.75 & 0.25 & -0.25 & 0.5 & 0 \\ & -0.25 & 0 & 0.5 & 0 \\ && -0.5 & 0.25 & 0.25 \\ &&& -0.5 & 0.25 \\ &&&& -0.25 \end{bmatrix} \end{equation}
Solution (Attempts)
According to the text, if I want to iteratively solve for $V_i$, I just have to repeat the following calculation until it converges, setting the unknown values to $0$ initially:
\begin{equation} V_k = - \frac{1}{C_{kk}} \sum^{n}_{i = 1, i \neq k} V_i C_{ki} \end{equation}
Attempt 1
However, when I do this, the values diverge:
| Iteration | V3 | V4 | V5 |
|-----------+-------------+-------------+-------------|
| 0 | 0 | 0 | 0 |
| 1 | -50 | 200 | 0 |
| 2 | 50 | 125 | 150 |
| 3 | 87.5 | 300 | 175 |
| 4 | 187.5 | 331.25 | 387.5 |
The equations have some of the right properties (non-adjacent nodes don't directly contribute to the value of a node in the iterative equations, the matrix is symmetric), but even going back and correcting for arithmetic errors, the divergent behavior remains the same.
Attempt 2
On the other hand, starting with something like
| Node | Coordinate | Value (V_i) |
|------+------------+-------------|
| 1 | 1,1 | 100 |
| 2 | 0,0 | * |
| 3 | 2,1 | 100 |
| 4 | 1,0 | 0 |
| 5 | 3,0 | * |
results in immediate convergence.
Is there something I can look for instead of repeating calculations?