I did this quick test in 2d for a Poisson problem https://github.com/cpraveen/fembook/blob/master/deal.II/ex04/demo.cc <https://github.com/cpraveen/fembook/blob/master/deal.II/ex04/demo.cc> (Change quadrature rules in this code as indicated below)
degree=1 assembly using QGauss(2) error computed using QGauss(2) cells dofs L2 H1seminorm 1024 1089 1.606e-03 - 2.517e-01 - 4096 4225 4.015e-04 2.00 1.259e-01 1.00 16384 16641 1.004e-04 2.00 6.295e-02 1.00 65536 66049 2.510e-05 2.00 3.148e-02 1.00 262144 263169 6.275e-06 2.00 1.574e-02 1.00 We just observe the standard convergence rates, does not indicate superconvergence. The following two also yield standard convergence rates degree=1 assembly using QGauss(2) error computed using QGaussLobatto(2) degree=1 assembly using QGaussLobatto(2) error computed using QGaussLobatto(2) This indicates there is no superconvergence at the mesh vertices. (In all cases above, the matrix is exactly assembled.) Best praveen > On 07-Jul-2020, at 8:53 AM, Wolfgang Bangerth <[email protected]> wrote: > > >> 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. > > -- > ------------------------------------------------------------------------ > Wolfgang Bangerth email: [email protected] > www: http://www.math.colostate.edu/~bangerth/ > > -- > The deal.II project is located at http://www.dealii.org/ > For mailing list/forum options, see > https://groups.google.com/d/forum/dealii?hl=en > --- You received this message because you are subscribed to the Google Groups > "deal.II User Group" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected]. > To view this discussion on the web visit > https://groups.google.com/d/msgid/dealii/d07ed649-dc11-53e4-7e48-3f348f97caf2%40colostate.edu. -- The deal.II project is located at http://www.dealii.org/ For mailing list/forum options, see https://groups.google.com/d/forum/dealii?hl=en --- You received this message because you are subscribed to the Google Groups "deal.II User Group" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/dealii/977AEB90-82E0-4149-99D3-CDEC501B3DCE%40gmail.com.
