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You can see why the sum of differences is giving a second derivative if you look at the derivative of a one-dimensional function $f$. The derivative of $f$ at $x_0$ is $$\frac{df}{dx}(x_0) \approx \frac{f(x_0 + dx) - f(x_0)}{dx}$$ The second derivative is then \begin{eqnarray*} \frac{d^2f}{dx^2}(x_0) &\approx& \frac{df/dx(x_0) - df/dx(x_0-dx)}{...
As you pointed out, the problem here is the discretization/subdivision of the mesh. If your mesh was made of quadrilaterals instead of triangles, the obvious subdivision strategy would be to split each quadliteral into four equally sized smaller quadliterals: $\hspace{2cm}$ $\hspace{2cm}$ For any two points $P_1$ and $P_2$, Dijkstra's algorithm would yield ...