Timeline for Dual basis of Lagrange nodal variables in $R^d$
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 13, 2016 at 0:09 | vote | accept | user177196 | ||
| Dec 12, 2016 at 4:54 | answer | added | VorKir | timeline score: 0 | |
| Dec 12, 2016 at 4:45 | comment | added | VorKir | In fact, no, I am sure there is a nice way to compute all this but now can think only of a brute force one. $\int_T \lambda_1^2 = \int_T x_1^2 dx = \int_0^1 x_1^2 V_{d-1}(1-x_1) dx_1$ where $V_k(r)$ is the volume of $k$-dimensional canonical simplex with side lentghs equal to $r$. So, I guess $V_k(r) = r^k V_k(1) = r^k / k!$. | |
| Dec 12, 2016 at 1:48 | comment | added | user177196 | @VorKir: I don't think mine above is correct (I plugged in and did not get $\delta_{ij}$ as I wish to), so that's why I asked because I hope you could help with the specific mechanism.... I have not figured out where it went wrong though, both for $\int_{T} \lambda^2_i dx$ and for $\int_{T} \lambda_i\lambda_j dx$ Please help if you could, as I don't think these calculations would take you more than 5-7 mins... | |
| Dec 12, 2016 at 1:45 | comment | added | VorKir | I am a bit too lazy to check the calculations. But either you are correct (and then your values of the integrals should satisfy the system with $\lambda_j$ and $\lambda_j^*$) or you are not and then they will not satisfy the equations. | |
| Dec 11, 2016 at 2:26 | comment | added | user177196 | So in $d$-case, for $i\neq j$, is $\int_{T} \lambda_{i}(x)\lambda_{j}(x) dx = d!J\times 1/(d(d+1)) = (d-1)!|T|/(d+1)$? And is $\int_{T} \lambda_{i}^2 dx = (d-1)!|T|/(2(d+1))?$ I searched the Wiki, and the volume of reference $d-dimensional$ hypersimplex is $\frac{1}{d!}$, so from your hint above, I get $J= Td!$. Is this correct? | |
| Dec 11, 2016 at 0:02 | comment | added | VorKir | $dx^{hat} = dx_1^{hat} dx_2^{hat} ..$ as a definition of the d-dimensional differential as in calculus, like a infinitesimal volume in d-dimensions, or I don't know how to put it. | |
| Dec 10, 2016 at 23:58 | comment | added | VorKir | As for the RHS, I was thinking about the first system - with i = 1, j = 1,2,3, so the Cronecker delta will give (1,0,0)^t in the RHS. | |
| Dec 10, 2016 at 23:57 | comment | added | VorKir | Actually, after change of variables you get the Jacobian $J$, which in 2D is equal to $2 area|T|$ for the reference triangle as described in the reference. In d-case, for an affine transform you will get area $T = J * $ volume of reference d-simplex. | |
| Dec 10, 2016 at 6:12 | comment | added | user177196 | @VorKir: I don't see how he got $dx^{hat} = dx^{hat}_1dx^{hat}_2$ (i.e, the 3rd equality when compute $\int_{T_k} \lambda^{2}_2(x)dx$. Could you kindly explain here if you got this step? | |
| Dec 10, 2016 at 6:11 | comment | added | user177196 | @VorKir: thank you so much for your help. But why for $d=2$, you got $RHS = (1,0,0)$? So for the general case with arbitrary $d$, is the area of a $d$-dimensional reference triangle $T$ constructed in a similar fashion as $T^{hat}_k$ on page 158 equal to $\frac{J}{2}$ still? I am asking if I could compute $\int_{T_k} \lambda_{1}(x) dx = \int_{T^{hat}_k} \lambda^{hat}_{1}(x^{hat}) J dx = 2area|T|\int_{0}^{1} \lambda^{hat}_{1} \int_{0}^{1}....\int_{0}^{1-x^{hat}_{1}-x^{hat}_2-...-x^{hat}_{d-1}} dx^{hat}_2dx^{hat}_3...dx^{hat}_{d-1}$? | |
| Dec 10, 2016 at 0:09 | comment | added | VorKir | Or, you can google it as me and find, e.g., the computed integrals ljll.math.upmc.fr/~ledret/M1English/M1ApproxPDE_Chapter5-2.pdf, page 158, | |
| Dec 10, 2016 at 0:06 | comment | added | VorKir | You can form a system for combination of integrals. E.g., the first system is for integrals of the products of lambda_1 and others lambda(_1,2,3) with righthand side (1,0,0) in case d=2. Then you can solve the system and find what are the values of integral lambda_i* lambda_j in terms of area of T. Then you need only to check that this is true by direct computations maybe. At least you can give it a try I guess. | |
| Dec 9, 2016 at 5:39 | comment | added | user177196 | Nobody wants to help me with this difficult problem? | |
| Dec 6, 2016 at 19:56 | history | asked | user177196 | CC BY-SA 3.0 |