In step-7
https://www.dealii.org/current/doxygen/deal.II/step_7.html
<https://nam01.safelinks.protection.outlook.com/?url=https%3A%2F%2Fwww.dealii.org%2Fcurrent%2Fdoxygen%2Fdeal.II%2Fstep_7.html&data=02%7C01%7CWolfgang.Bangerth%40colostate.edu%7C95a7523f571844a4dbc308d8209dbf1a%7Cafb58802ff7a4bb1ab21367ff2ecfc8b%7C0%7C0%7C637295209184273315&sdata=PbscIaF1YgoA6TOFydENsrn79iERjXLdAeV9g7b5u94%3D&reserved=0>
under section on "Verification of correctness”, there is this statement
*(e.g., for linear elements, do not use the QGauss(2) quadrature formula)*
because solution may exhibit superconvergence at the QGauss(2) points.
If we solve
-u’’ = 16*pi^2*sin(4*pi*x) in (0,1)
u(0) = 0, u(1) = 1
using 8 linear elements and QGauss(2) for quadrature.
The error is very small at the vertices of the mesh, not at the QGauss(2)
points.
Can you look into this issue, is the comment in the documentation wrong,
perhaps it should say *do not use QGaussLobatto(2)* ? Or is there some issue I
am missing here ?
Ah, very interesting question! You're right that in some situations -- the
Laplace equation in 1d specifically -- the superconvergence points are in fact
the vertices of the cells.
But that's not true in 2d/3d. There, at least the recollection I have from
when I learned about this many years ago, the superconvergence points are
indeed the Gauss points. Want to try that out as well in a small experiment?
Say take a 16x16 mesh, and plot both solution and discrete solution in a part
of the domain well away from the boundary, and see where the two seem to
intersect.
(As always, we're always happy to improve the documentation. Clearly, what I
said in step-7 is not the complete truth and ought to be improved, but I'd
rather we check what we say before coming up with a better description :-) )
Cheers
W.
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Wolfgang Bangerth email: [email protected]
www: http://www.math.colostate.edu/~bangerth/
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