Dear all,
I would like to address a problem related to the interpretation of the
results of a convergence test. I am performing a temporal convergence test
to test a solver for advection-diffusion problems. The test is based on a
manufactured solution on a unit square domain. The Peclet number of the
test is equal to 100 which is a rather modest value.
I obtain the following two convergence tables for linear and quadratic
elements, FE_Q<2>(1) and FE_Q<2>(2), respectively.
Convergence table for linear elements
============================================
dt dofs hmax L2
1.00e-01 16641 1.10e-02 9.057479e-03 -
5.00e-02 16641 1.10e-02 2.764636e-03 -1.71
2.50e-02 16641 1.10e-02 7.512931e-04 -1.88
1.25e-02 16641 1.10e-02 1.940537e-04 -1.95
6.25e-03 16641 1.10e-02 4.835196e-05 -2.00
3.12e-03 16641 1.10e-02 1.118790e-05 -2.11
1.56e-03 16641 1.10e-02 2.037621e-06 -2.46
7.81e-04 16641 1.10e-02 1.227090e-06 -0.73
3.91e-04 16641 1.10e-02 1.609307e-06 0.39
1.95e-04 16641 1.10e-02 1.724263e-06 0.10
Convergence table for quadratic elements
============================================
dt dofs hmax L2
1.00e-01 66049 1.10e-02 9.057858e-03 -
5.00e-02 66049 1.10e-02 2.765966e-03 -1.71
2.50e-02 66049 1.10e-02 7.526724e-04 -1.88
1.25e-02 66049 1.10e-02 1.954472e-04 -1.95
6.25e-03 66049 1.10e-02 4.974689e-05 -1.97
3.12e-03 66049 1.10e-02 1.255001e-05 -1.99
1.56e-03 66049 1.10e-02 3.156146e-06 -1.99
7.81e-04 66049 1.10e-02 7.957238e-07 -1.99
3.91e-04 66049 1.10e-02 2.035262e-07 -1.97
1.95e-04 66049 1.10e-02 5.312767e-08 -1.94
What bothers me is that in case of linear elements the error is increasing
again when the time step becomes very small (positive rate). For quadratic
elements it is monotonously decreasing, which is what I expect.
I think that in case of linear elements the error becomes dominated by the
spatial discretization error. The spatial discretization error should be
the lower bound of the "total error" and it is smaller when quadratic
elements is used instead of linear ones. On that basis, I can understand
that a further decrease of the error beyond a certain lower bound is not
possible, but I cannot understand the subsequent increase in case of linear
elements. Have you ever experienced such problem and a possible
explanation? Do you think that I can trust my program based on the result?
I also thought about two other sources of error: the quadrature error and
the error when solving the linear system iteratively. I tried to reduce the
second one as much as possible, but this was not successful. Do you have
suggestions how to find additional sources of error?
Best wishes,
Sebastian
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