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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