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.

翻译：双一致性是发展稳定的DWR误差估计以实现面向目标网格自适应的重要问题。本文基于牛顿-GMG框架深入研究了稳态欧拉方程中的这一问题。理论上，采用Petrov-Galerkin方案重新描述了数值框架，并以此刻画了双一致性。讨论了边界修正技术以保持牛顿-GMG框架内的双一致性。数值上，提出了一种几何多重网格方法求解对偶问题，并设计了正则项以保证迭代收敛。数值实验表明该方法具有以下特征：i) 对于不同构型的问题，关注量的数值收敛能平稳实现；ii) 与双不一致方法相比，本文提出的双一致DWR方法在精确计算关注量时可显著节省网格数量。