In addition what Wolfgang said:
1. It would be indeed interesting to
see whether neglecting z_h really yields
the same error, the same effectivity indices,
and/or the same mesh.
2. Going back to your initial questions:
Inserting z_h is the key when classical
a posteriori bounds in terms of the mesh size h
are of interest.
The final goal is usually to obtain an error estimate
in terms of the mesh size h in order to
quantify the order of convergence of your scheme.
For this reason you need to insert z_h such that
|| z - z_h||
to apply interpolation estimates that give you
some h^{a} on the right hand side:
|| z-z_h || = O(h^a)
with the order a.
3. In practice you have indeed different choices
how to evaluate J(u) - J(u_h).
Also some people do not integrate back
into the strong form and work
with a weak form of the error estimator, which
has the advantage that no second-order operators
and partial integration needs to be applied.
From this point of view, it may be that neglecting
z_h could work in practice. But as said above,
a careful study for some model problems
would be useful.
Best Thomas W.
--
++--------------------------------------------++
Prof. Dr. Thomas Wick
Institut für Angewandte Mathematik (IfAM)
Leibniz Universität Hannover
Welfengarten 1
30167 Hannover, Germany
Tel.: +49 511 762 3360
Email: thomas.w...@ifam.uni-hannover.de
www: http://www.ifam.uni-hannover.de/wick
www: http://www.cmap.polytechnique.fr/~wick/
++--------------------------------------------++
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On 03/10/2018 11:03 AM, Wolfgang Bangerth wrote:
On 03/06/2018 08:40 AM, 曾元圆 wrote:
Now I understand why we need to rewrite the error formula on a cell
as residual times dual weight. But I'm still a little confused with
the reason why we must introduce z_h.
Just as you mentioned, if we introduce z_h, then z-z_h is a quantity
that is only large where the dual solution is rough. But why do we
need to care about the accuracy of z here? I think the only thing we
need to care about is the value of z on that cell, because z is a
quantity that represents how important the residual on that cell is.
No. z tells you how important the *locally generated error is for the
global error functional*. (That is because z is the Green's function
associated with your error functional.) But you don't have the local
error. All you have is the local residual.
My understanding is: now the dual_weight z-z_h does not only
represent how important the residual on a certain cell is, but also
tells us some information about how good the dual solution on that
cell is. But another problem is, does z-z_h still has the same
tendency as z?
Almost. Think of it as z-phi_h where you can choose phi_h as you want.
For example, on each cell you can think of choosing phi_h so that it
cancels the constant and linear term of the Taylor expansion of z.
Then z-z_h would contain the quadratic and higher order Taylor terms,
i.e. something like z''*(x-x0)^2 where x0 can be chosen as a point on
the cell.
If not, how z-z_h can represent the importance of a certan cell as z
can?
I'm not sure if my understanding is correct. I tried to run the code
using only z as dual_weights, and I found the result almost the same
as that using z-z_h.
Nice idea to try this out. Do you get "almost the same" overall error
estimate, or "almost the same" mesh?
I think all of these are good questions to ask. Although I have worked
on this for a long time, I can not actually give you a particularly
good answer for all of this. I am sure others who are more versed in
the theory of errors, residuals, etc could tell you the precise reason
for why it is in fact necessary to subtract z_h. The best I can say is
that that's the way I've always seen it done, and while I have a vague
idea why that is so (see above), I can't say that I can describe it
well enough to explain it.
Finally, I am certainly glad to submit patches to deal.II and make my
own contribution. But I didn't fork deal.ii on my github account yet,
and this is relatively a small issue, so I will be glad if you can do
it for the moment.
OK, I will take care of this then.
Best
W.
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