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I found this formula in an engineering textbook (image processing). It is an approximation of the Laplacian on flat space $\mathbb{R}^2$.

\begin{eqnarray*} \nabla^2 f &\approx& -20 f(\vec{x}) + 4 \big( f( \vec{x} + \epsilon \,\mathbf{i} ) + f( \vec{x} + \epsilon \,\mathbf{j} ) + f( \vec{x} - \epsilon \,\mathbf{i} ) + f( \vec{x} - \epsilon \,\mathbf{j} )\big) \\ &+& \big( f(\vec{x}+ \epsilon (\mathbf{i}+ \mathbf{j})\big) + \big( f(\vec{x}+ \epsilon (\mathbf{i}- \mathbf{j})\big) + \big( f(\vec{x}+ \epsilon (-\mathbf{i}+ \mathbf{j})\big) + \big( f(\vec{x}+ \epsilon (-\mathbf{i}- \mathbf{j})\big) \end{eqnarray*}

Sorry I have no succinct way of writing this. Typically one uses a Stencil:

$$ \frac{1}{6\epsilon^2} \left[ \begin{array}{ccc} 1 & 4 & 1 \\ 4 & -20 & 4 \\ 1 & 4 & 1 \end{array} \right] $$

I suspect this falls out of Euler MacLaurin formula. But he also estimate the error term. More precisely if you call the right hand side $L$ he says that:

$$ L = \nabla^2 + \frac{\epsilon^2}{12} \nabla^2( \nabla^2) + O(\epsilon)^4 $$

Can one formalize such an estimate using Sobolev norms?

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  • $\begingroup$ @reuns that's a good estimate as any... these engineers * really * wanted to know how bad the Stencil was. $\endgroup$
    – john mangual
    Commented Dec 11, 2017 at 0:55
  • $\begingroup$ $| \frac{f(x-\epsilon)-2 f(x)+f(x+\epsilon)}{\epsilon^2}-f''(x)| = |\frac1{\epsilon^2}\int_0^{\epsilon}\int_{x-\epsilon+t}^{x+t} (f''(\tau)-f''(x))d\tau dt|$ $\endgroup$
    – reuns
    Commented Dec 11, 2017 at 1:00
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    $\begingroup$ using Sobolev norms Which ones? (how much smoothness do you allow, what $p$, etc.) $\endgroup$
    – fedja
    Commented Dec 11, 2017 at 4:57
  • $\begingroup$ What about a pointwise estimate using Taylor series of each term (assuming continuous differentiability of $f$)? I wouldn't suspect anything more fancy that this. $\endgroup$
    – Dirk
    Commented Dec 11, 2017 at 5:46
  • $\begingroup$ @fedja correct. The author of this textbook does not know any analysis. But I found his estimate very interesting $\endgroup$
    – john mangual
    Commented Dec 11, 2017 at 13:16

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