Adaptive solution summary is suspicious

[Former user]

The following is the output of demo_auto_adaptive_poisson.py on a fresh install

 Level  |  functional_value  error_estimate  tolerance  num_cells  num_dofs
  --------------------------------------------------------------------------
  0      |          0.121629        0.001179      1e-05        128        81
  1      |          0.126801     0.000623548      1e-05        268       158
  2      |          0.124812     0.000335811      1e-05        493       273
  3      |          0.124488     0.000219046      1e-05       1051       566
  4      |          0.125291     8.35258e-05      1e-05       1839       969
  5      |           0.12497     7.26539e-05      1e-05       3695      1917
  6      |           0.12524      2.5749e-05      1e-05       6552      3366
  7      |          0.125153     2.01576e-05      1e-05      11869      6057
  8      |          0.125229     8.81699e-06      1e-05      22424     11387

To my understanding of the error_estimate values, they give the size of the neighborhood of the corresponding functional_value in which the real functional value belongs, but then I would expect these neighborhoods to include each other, which is not happening. For example, the first line reads 0.121629 +/- 0.001179 and the last one reads 0.125229 +/- 8.81699e-06 but these neighborhoods are disjunct.

Am I misinterpreting the output?

[Marie Elisabeth Rognes]

The error estimate provided by the auto-adaptive algorithm is an estimate and not guaranteed to be an upper bound on the error. The behaviour you observe illustrates this.

[Marie Elisabeth Rognes]

This is a feature not a bug.

[Former user]

I don't get the point of having an error estimate that is an estimate in the sense that it gives no information on the real error. What should the user do with that number?

[Jan Blechta]

I don't get the point of having an error estimate that is an estimate in the sense that it gives no information on the real error. What should the user do with that number?

There are more competing theories in a posteriori error estimation theories. Some of them (like some of residual methods, equilibrated flux reconstruction methods, etc.) provide guaranteed upper bound on error (measured in a particular way), while others (like goal-oriented error estimation, originating in the works of Rannacher, Becker, and others) do not, in general, guarantee the error to be bounded by the estimate. Still there is some belief in mathematical community that this is still useful.