Timeline for Discrete approximations to $\nabla^2$
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 7, 2018 at 15:38 | comment | added | john mangual | @AlexGavrilov you are showing there's a problem with the textbook (whiich was written for engineers) | |
| Jan 7, 2018 at 15:07 | comment | added | Alex Gavrilov | One obvious observation: you need quite a smooth function for this. Even for $f\in H^5$ the formula does not make sense because then $\Delta^2f\in H^1$ and this function is not necessarily continuous. For $f\in H^6$ the answer is clearly yes, except it may be a little bothersome. | |
| Dec 11, 2017 at 13:16 | comment | added | john mangual | @fedja correct. The author of this textbook does not know any analysis. But I found his estimate very interesting | |
| Dec 11, 2017 at 5:46 | comment | added | Dirk | 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. | |
| Dec 11, 2017 at 4:57 | comment | added | fedja | using Sobolev norms Which ones? (how much smoothness do you allow, what $p$, etc.) | |
| Dec 11, 2017 at 1:00 | comment | added | reuns | $| \frac{f(x-\epsilon)-2 f(x)+f(x+\epsilon)}{\epsilon^2}-f''(x)| = |\frac1{\epsilon^2}\int_0^{\epsilon}\int_{x-\epsilon+t}^{x+t} (f''(\tau)-f''(x))d\tau dt|$ | |
| Dec 11, 2017 at 0:55 | comment | added | john mangual | @reuns that's a good estimate as any... these engineers * really * wanted to know how bad the Stencil was. | |
| Dec 11, 2017 at 0:40 | history | asked | john mangual | CC BY-SA 3.0 |