again, i would appreciate your kind help.
I would like to solve the adjoint equation of my following problem to get to the formoptimisation.
whereby $$\phi$$ is the potential, n is the outer normal and $$I_2(t)$$ is a current.
Defining V as
and multiplying with the test function v and doing the integration yields for the weak formulation
Now we are applying a little trick by saying v should be $$\phi$$ which yields now
This would give an optimal control problem with the energy-functional like
under the constrains of the potential equation.
Ok. Now i approximate through the approach functions for the finite-dimensional problem like
and getting with $$\gamma$$ be a constant
where one can see the stiffness matrix.
I believe now my problem starts and i am really sorry if i am asking now stupid questions.
In matrix form it would be (maybe)
and for the adjoint equation i would invert and transpose the stiffness matrix K. But then i only have stupid stuff on my LHS, or? Can someone tell me what i am doing wrong, please?