Thank you so much Bangerth!
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.
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? 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
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
在 2018年3月4日星期日 UTC+8上午1:42:27，Wolfgang Bangerth写道：
> On 03/02/2018 03:36 AM, 曾元圆 wrote:
> > Hi, nowadays I'm learning about the goal oriented error estimator and I
> > read the tutorial of step-14
> > (http://www.dealii.org/developer/doxygen/deal.II/step_14.html.) But I'm
> > confused about some problems and hope you can help me with these: 1.
> > deriving the error with respect to the functional, why must we change
> > J(e)=a(e,z) to J(e)=a(e,z-z_h) ? I know the dual solution z must be
> > approximated in a richer space than the primal solution, otherwise J(e)
> > will be 0. But why not just solve the dual problem in a richer space
> > without subtracting its interpolation to the primal space? I didn't see
> > necessity to introduce z_h into the formula.
> You are correct: the values you get from both of these formulas are
> the same. So it is not *necessary* to introduce z_h if you are interested
> computing the *error*.
> We introduce the interpolant because z-z_h is a quantity that is only
> where the dual solution is rough, where z may be large also in other
> Doing so ensures that the error estimator is *localized*: It is large
> where the primal and dual solutions are rough, i.e., where we expect the
> to be caused. As mentioned above, if you sum the contributions of all
> you will get the same value whether you introduce z_h or not, but the
> contribution of each cell is going to be different. If you want the
> contributions of each cell to serve as a good mesh refinement criterion,
> it turns out that you need to introduce z_h.
> > 2. Why must we change J(e)=∑(f+△ uh,z-zh)-(∂uh,z-zh) to J(e)=∑(f+△
> > uh,z-zh)-(1/2[∂uh],z-zh)? Is it just an implementation consideration for
> > saving computational effort?
> For the same kind of reason. For a smooth (primal) solution, the term
> may be large because the normal derivative of the primal solution may
> be what it is -- think of, for example, a linear exact solution u that can
> perfectly approximated by u_h. So if you leave this term as is, this would
> suggest that the error on this cell is large. But that's wrong -- the
> error is
> actually quite small because you can approximate linear functions well.
> On the other hand, if you do the rewrite (which again leaves the *sum*
> all cells the same, but change the contributions of each cell), then
> is going to be small because while the normal derivative of u_h may be
> the *jump* of the normal derivative is small if the solution is linear or
> nearly so.
> Another way of seeing this is to think of both of the terms in J(e) as
> residual times dual weight.
> The residuals here are f+△ uh and 1/2[∂uh]. You want to define these
> in such a way that they are zero for the exact solution. That is true for
> these two residuals: f+△ u is zero because 'u' satisfies the equation, and
> 1/2[∂u] is zero because the solutions of the Laplace equation have
> gradients (if f is smooth enough).
> On the other hand, the term ∂u is not zero, even for the exact solution.
> > Can this kind of rewriting be generally adopted in other kind of
> > problems(e.g in advection problem where the face integrals in J(e)
> > on the upstream information)?
> Wolfgang Bangerth email: bang...@colostate.edu
> www: http://www.math.colostate.edu/~bangerth/
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