I am trying to derive the adjoint Poisson equation for the following problem to find the sensitivity of an objective function with respect to a decision variable, but I get stuck in the middle of the way. Here is my derivation.
The objective function is:
$$\min_{a} J=\int_{\Omega} cu(a,x)d\Omega$$
in which $c$ is a constant, $a$ is the decision variable, and $u(a,x)$ is a solution to the following Poisson equation.
$$\nabla \cdot (a \nabla u) - f = 0$$
In PDE constraint (Poisson equation), $f$ is constant and $a$ is a function of position. The Poisson equation is defined on a domain $\Omega$ which is bounded by the boundary $\partial \Omega$. The boundary has two parts, $\partial \Omega = \partial \Omega_1 \cup \partial \Omega_2$. The boundary conditions for the Poisson equation is as follows:
$$u=0 \, \text{on} \, \partial \Omega_1$$ $$a \nabla u \cdot \hat{n} = h(u_{\partial \Omega_2}-u_{\mathrm{bulk}}) \, \text{on} \, \partial \Omega_2$$
where $\hat{n}$ is a unit vector normal to the boundary toward outside of the domain, $u_{\partial \Omega_2}$ is value of $u$ on boundary $\partial \Omega_2$, $u_{\mathrm{bulk}}$ is a constant value, and $h$ is a constant convection coefficient.
To derive the adjoint problem, the Lagrangian is expressed as:
$$L = J + \int_{\Omega} \lambda \left[ \nabla \cdot (a \nabla u) - f \right] d\Omega = \int_{\Omega} cud\Omega + \int_{\Omega} \lambda \left[ \nabla \cdot (a \nabla u) - f \right] d\Omega$$
The aim is to calculate sensitivity of $J$ with respect to $a$ ($\frac{\delta J}{\delta a}$). Hence, $\delta J$ is given by:
$$\delta J = \delta L = \int_{\Omega} c\delta u d\Omega + \int_{\Omega} \lambda \left[ \nabla \cdot (\delta a \nabla u) + \nabla \cdot (a \nabla \delta u) \right] d\Omega$$
To find the term $\int_{\Omega} \lambda \nabla \cdot (a \nabla \delta u) d\Omega$, I performed integration by parts twice.
The first integration by parts:
$$\int_{\Omega} \lambda \nabla \cdot (a \nabla \delta u) d\Omega = \int_{\partial\Omega_1} \lambda a \nabla \delta u \cdot \hat{n} dS + \int_{\partial\Omega_2} \lambda a \nabla \delta u \cdot \hat{n} dS - \int_{\Omega} a \nabla \delta u \cdot \nabla \lambda d\Omega $$
I set the the value of $\lambda$ on $\partial\Omega_1$ to be equal to zero. Hence, the first term on RHS of the equation above is eliminated and the equation can be expressed as:
$$\int_{\Omega} \lambda \nabla \cdot (a \nabla \delta u) d\Omega = \int_{\partial\Omega_2} \lambda a \nabla \delta u \cdot \hat{n} dS - \int_{\Omega} a \nabla \delta u \cdot \nabla \lambda d\Omega $$
Now, I perform the second integration by parts:
$$\begin{aligned} \int_{\Omega} \lambda \nabla \cdot (a \nabla \delta u) d\Omega = \int_{\partial\Omega_2} \lambda a \nabla \delta u \cdot \hat{n} dS - \int_{\partial\Omega_1} a \delta u \nabla \lambda \cdot \hat{n} dS - \int_{\partial\Omega_2} a \delta u \nabla \lambda \cdot \hat{n} dS \\ + \int_{\Omega} \delta u \nabla \cdot (a \nabla \lambda) d\Omega \end{aligned}$$
Since the value of $u$ is fixed on $\partial\Omega_1$, the second term on RHS on this equation can be omitted. The final result of integration by part is as follows:
$$\int_{\Omega} \lambda \nabla \cdot (a \nabla \delta u) d\Omega = \int_{\partial\Omega_2} \lambda a \nabla \delta u \cdot \hat{n} dS - \int_{\partial\Omega_2} a \delta u \nabla \lambda \cdot \hat{n} dS + \int_{\Omega} \delta u \nabla \cdot (a \nabla \lambda) d\Omega $$
Now, $\delta J$ could be written as:
$$\begin{aligned} \delta J = \int_{\Omega} c\delta u d\Omega + \int_{\Omega} \lambda \nabla \cdot (\delta a \nabla u) d\Omega + \int_{\partial\Omega_2} \lambda a \nabla \delta u \cdot \hat{n} dS - \int_{\partial\Omega_2} a \delta u \nabla \lambda \cdot \hat{n} dS \\ + \int_{\Omega} \delta u \nabla \cdot (a \nabla \lambda) d\Omega \end{aligned}$$
This is the point that I do not know how to proceed further. I do not know how I should set the boundary condition of $\lambda$ on $\partial\Omega_2$. Basically, I do not know if I should eliminate the third or fourth integral in the last equation. Besides, after setting the boundary condition of $\lambda$ on $\partial\Omega_2$ and eliminating either of terms, I do not know how to treat with the other integral term in calculation of $\frac{\delta J}{\delta a}$.
I appreciate any help or resources.
Thanks