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For example, consider a 1D data $\begin{bmatrix} a\\ b\\ c \end{bmatrix}$.

I can represent forward difference operator as matrix

$\nabla_n = \begin{bmatrix} -1&1& 0\\ 0&-1&1\\ 0&0& 0\end{bmatrix}$ (neumann boundary condition) and

$\nabla_p = \begin{bmatrix} -1&1& 0\\ 0&-1&1\\ 1&0& -1\end{bmatrix}$ (periodic boundary condition).

Then we have

$\nabla_n^T \nabla_n=\begin{bmatrix} 1&-1& 0\\ -1&2&-1\\ 0&-1& 1\end{bmatrix}=-\Delta_n$ (neumann laplacian) and

$\nabla_p^T \nabla_p=\begin{bmatrix} 2&-1& -1\\ -1&2&-1\\ -1&-1& 2\end{bmatrix}=-\Delta_p$ (periodic laplacian).

My question is that I don't know how to represent Dirichlet boundary condition. I thought

$\nabla_d = \begin{bmatrix} -1&1& 0\\ 0&-1&1\\ 0&0& -1\end{bmatrix}$,

but then

$\nabla_d^T \nabla_d=\begin{bmatrix} 1&-1& 0\\ -1&2&-1\\ 0&-1& 2\end{bmatrix} \neq -\Delta_p = \begin{bmatrix} 2&-1& 0\\ -1&2&-1\\ 0&-1& 2\end{bmatrix}$.

How can I represent Dirichlet case?

Nils Werner
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Gobi
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  • Great question, couldn't solve it so far. It appears somehow operator with the Dirichlet boundary condition is special. I found someone asked a similar question here (http://math.stackexchange.com/questions/1586392/discretization-of-forward-gradient-and-backward-divergence-with-dirichlet-or-n), and there is no answer. – Taozi Nov 29 '16 at 21:12
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    Why should the product of the difference operator matrix with the transpose of itself give an approximation of the laplacian? – Bananach Dec 01 '16 at 22:36

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