  The dual consistency is an important issue in developing stable DWR error
estimation towards the goal-oriented mesh adaptivity. In this paper, such an
issue is studied in depth based on a Newton-GMG framework for the steady Euler
equations. Theoretically, the numerical framework is redescribed using the
Petrov-Galerkin scheme, based on which the dual consistency is depicted. A
boundary modification technique is discussed for preserving the dual
consistency within the Newton-GMG framework. Numerically, a geometrical
multigrid is proposed for solving the dual problem, and a regularization term
is designed to guarantee the convergence of the iteration. The following
features of our method can be observed from numerical experiments, i). a stable
numerical convergence of the quantity of interest can be obtained smoothly for
problems with different configurations, and ii). towards accurate calculation
of quantity of interest, mesh grids can be saved significantly using the
proposed dual-consistent DWR method, compared with the dual-inconsistent one.
