Consider the following linear, scalar advection-diffusion equation for the primal solution ${\displaystyle u({\vec {x}})}$ , in the domain ${\displaystyle \Omega }$  with Dirichlet boundary conditions:

${\displaystyle {\begin{aligned}\nabla \cdot \left({\vec {c}}u-\mu \nabla u\right)&=f,\qquad {\vec {x}}\in \Omega ,\\u&=b,\qquad {\vec {x}}\in \partial \Omega .\end{aligned}}}$ 

Let the output of interest be the following linear functional:

${\displaystyle J(u)=\int _{\Omega }gu\ dV.}$ 

Derive the weak form by multiplying the primal equation with a weighting function ${\displaystyle w({\vec {x}})}$  and performing integration by parts:

${\displaystyle {\begin{aligned}B(u,w)&=L(w),\end{aligned}}}$ 

where,

${\displaystyle {\begin{aligned}B(u,w)&=\int _{\Omega }w\nabla \cdot \left({\vec {c}}u-\mu \nabla u\right)dV\\&=\int _{\partial \Omega }w\left({\vec {c}}u-\mu \nabla u\right)\cdot {\vec {n}}dA-\int _{\Omega }\nabla w\cdot \left({\vec {c}}u-\mu \nabla u\right)dV,\qquad {\text{(Integration by parts)}}\\L(w)&=\int _{\Omega }wf\ dV.\end{aligned}}}$ 

Then, consider an infinitesimal perturbation to ${\displaystyle L(w)}$  which produces an infinitesimal change in ${\displaystyle u}$  as follows:

${\displaystyle {\begin{aligned}B(u+u',w)&=L(w)+L'(w)\\B(u',w)&=L'(w).\end{aligned}}}$ 

Note that the solution perturbation ${\displaystyle u'}$  must vanish at the boundary, since the Dirichlet boundary condition does not admit variations on ${\displaystyle \partial \Omega }$ .

Using the weak form above and the definition of the adjoint ${\displaystyle \psi ({\vec {x}})}$  given below:

${\displaystyle {\begin{aligned}L'(\psi )&=J(u')\\B(u',\psi )&=J(u'),\end{aligned}}}$ 

we obtain:

${\displaystyle {\begin{aligned}\int _{\partial \Omega }\psi \left({\vec {c}}u'-\mu \nabla u'\right)\cdot {\vec {n}}dA-\int _{\Omega }\nabla \psi \cdot \left({\vec {c}}u'-\mu \nabla u'\right)dV&=\int _{\Omega }gu'\ dV.\end{aligned}}}$ 

Next, use integration by parts to transfer derivatives of ${\displaystyle u'}$  into derivatives of ${\displaystyle \psi }$ :

${\displaystyle {\begin{aligned}\int _{\partial \Omega }\psi \left({\vec {c}}u'-\mu \nabla u'\right)\cdot {\vec {n}}dA-\int _{\Omega }\nabla \psi \cdot \left({\vec {c}}u'-\mu \nabla u'\right)dV-\int _{\Omega }gu'\ dV&=0\\\int _{\partial \Omega }\psi \left({\vec {c}}u'-\mu \nabla u'\right)\cdot {\vec {n}}dA+\int _{\Omega }u'\left(-{\vec {c}}\cdot \nabla \psi \right)dV+\int _{\Omega }\nabla u'\cdot \left(\mu \nabla \psi \right)dV-\int _{\Omega }gu'\ dV&=0\\\int _{\partial \Omega }\psi \left({\vec {c}}u'-\mu \nabla u'\right)\cdot {\vec {n}}dA+\int _{\Omega }u'\left(-{\vec {c}}\cdot \nabla \psi \right)dV+\int _{\partial \Omega }u'\left(\mu \nabla \psi \right)\cdot {\vec {n}}dA-\int _{\Omega }u'\nabla \cdot \left(\mu \nabla \psi \right)dV-\int _{\Omega }gu'\ dV&=0\qquad {\text{(Repeating integration by parts on diffusion volume term)}}\\\int _{\Omega }u'\left[-{\vec {c}}\cdot \nabla \psi -\nabla \cdot \left(\mu \nabla \psi \right)-g\right]dV+\int _{\partial \Omega }\psi \left({\vec {c}}u'-\mu \nabla u'\right)\cdot {\vec {n}}dA+\int _{\partial \Omega }u'\left(\mu \nabla \psi \right)\cdot {\vec {n}}dA&=0.\end{aligned}}}$ 

The adjoint PDE and its boundary conditions can be deduced from the last equation above. Since ${\displaystyle u'}$  is generally non-zero within the domain ${\displaystyle \Omega }$ , it is required that ${\displaystyle \left[-{\vec {c}}\cdot \nabla \psi -\nabla \cdot \left(\mu \nabla \psi \right)-g\right]}$  be zero in ${\displaystyle \Omega }$ , in order for the volume term to vanish. Similarly, since the primal flux ${\displaystyle \left({\vec {c}}u'-\mu \nabla u'\right)\cdot {\vec {n}}}$  is generally non-zero at the boundary, we require ${\displaystyle \psi }$  to be zero there in order for the first boundary term to vanish. The second boundary term vanishes trivially since the primal boundary condition requires ${\displaystyle u'=0}$  at the boundary.

Therefore, the adjoint problem is given by:

${\displaystyle {\begin{aligned}-{\vec {c}}\cdot \nabla \psi -\nabla \cdot \left(\mu \nabla \psi \right)&=g,\qquad {\vec {x}}\in \Omega ,\\\psi &=0,\qquad {\vec {x}}\in \partial \Omega .\end{aligned}}}$ 

Note that the advection term reverses the sign of the convective velocity ${\displaystyle {\vec {c}}}$  in the adjoint equation, whereas the diffusion term remains self-adjoint.

• Adjoint state method
• Costate equations

• Jameson, Antony (1988). "Aerodynamic Design via Control Theory". Journal of Scientific Computing. 3 (3): 233–260. doi:10.1007/BF01061285. hdl:2060/19890004037. S2CID 7782485.