A note on consistency and adjointness for numerical schemes

This work deals with the consistency of nite di erence approximations. We investigate the relation between the consistency of a numerical scheme and the consistency of its adjoint. We exhibit examples of numerical schemes which are consistent with a (direct) equation and whose adjoint is not consistent with the adjoint equation. This undesirable feature appears in the application of the adjoint state technique which requires an adjointness relation to be satis ed. Therefore, the numerical scheme for the adjoint equation is determined by the choice of the numerical scheme on the direct equation. We conclude that in general consistency is not conserved by adjointness.