Wolfgang,
Thanks a lot for the response and for the suggestion to look at a simple 1d
example. I will work on this and indeed report back!
Kind regards,
-Sean
On Monday, September 29, 2025 at 2:51:40 PM UTC-4 Wolfgang Bangerth wrote:
>
> On 9/27/25 10:27, Sean Carney wrote:
> >
> > In implementing this finite difference check in deal.II, however, I only
> > observe a decrease in the error between (*) and (**) up to \epsilon
> > \approx 10^{-3}. After, that, the error increases. The details can be
> > seen the attached screenshot.
> >
> > For reference, I am using a (sparse) direct solver for solving both the
> > state and adjoint equations. The behavior is essentially the same if a
> > CG solver is used instead. I am using Q2 finite elements, but I observe
> > the same behavior for Q3 as well.
> >
> > Additionally, it is worth mentioning that in a sanity-check-test where
> > the objective function is modified to simply be:
> > J(z) = \alpha/2 \| z\|_2^2
> > so that the derivative is simply
> > J'(z) = \alpha z
> > (i.e. no adjoint calculation needed), the convergence behavior */is/*
> > what one would expect (i.e. the error decreases up to \epsilon ~ 1e-7).
> > So, the issue should be with the adjoint solve.
> >
> > *Can anyone point out a reason why I /shouldn't /expect the standard
> > behavior?* Or, perhaps there is an error in my adjoint calculation. I
> > tried to produce a light-weight, minimal-working-example. It is attached
> > here.
>
> Sean,
> I don't see a particular reason why you shouldn't get the expected
> convergence, but have also not spent as much time as you thinking this
> through. Does the discretization error not play a role in this?
> Presuambly you are comparing the FD gradient against an analytical one?
> Or against the gradient computed via the dual solution?
>
> My suggestion would be to make the problem simple. For example, let z
> use piecewise constants and u,p use piecewise quadratics. Then if you
> let u_D be a constant (perhaps zero), and if you work in 1d, the finite
> element approximation gives you the exact solution. In those cases, you
> can probably compute the exact solution of the problem on a piece of
> paper, along with what the exact gradient is. You should be able to
> compare both the adjoint-based gradient and the FD gradient against that.
>
> So these would be my steps to debug the issue -- please report back what
> you find!
>
> Best
> W.
>
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