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The continuous adjoint approach applied to the stabilized finite-element formulation of the incompressible Navier-Stokes equations

The finite element solution of the continuous adjoint to the Navier-Stokes equations requires stabilization similar to that required for the flow problem. This paper presents the stabilized adjoint equations and their boundary conditions, with the weak forms for the equations and boundary conditions. In addition to this, the weak form of the direct differentiation method is also presented. Finally, a test case using an S-shaped duct shows the agreement between the direct differentiation method and the adjoint method. Sensitivities and values of the adjoint solutions are given as validation data.