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.
--
------------------------------------------------------------------------
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].
For more options, visit https://groups.google.com/d/optout.
