An adjoint equation is a linear differential equation, usually derived from its primal equation using integration by parts. Gradient values with respect to a particular quantity of interest can be efficiently calculated by solving the adjoint equation. Methods based on solution of adjoint equations are used in wing shape optimization, fluid flow control and uncertainty quantification.

Example: Advection-Diffusion PDE

Consider the following linear, scalar advection-diffusion equation with Dirichlet boundary conditions:

Let the output of interest be the following linear functional:

Derive the weak form and performing integration by parts:

where,

as follows:

.

given below:

we obtain:

:

at the boundary.

Therefore, the adjoint problem is given by:

in the adjoint equation, whereas the diffusion term remains self-adjoint.

See also

• Adjoint state method

• Costate equations

References