The dual consistency, which is an important issue in developing dual-weighted residual error estimation towards the goal-oriented mesh adaptivity, is studied in this paper both theoretically and numerically. Based on the Newton-GMG solver, dual consistency had been discussed in detail to solve the steady Euler equations. Theoretically, based on the Petrov-Galerkin method, the primal and dual problems, as well as the dual consistency, are deeply studied. It is found that dual consistency is important both for error estimation and stable convergence rate for the quantity of interest. Numerically, through the boundary modification technique, dual consistency can be guaranteed for the problem with general configuration. The advantage of taking care of dual consistency on the Newton-GMG framework can be observed clearly from numerical experiments, in which an order of magnitude savings of mesh grids can be expected for calculating the quantity of interest, compared with the dual-inconsistent implementation. Besides, the convergence behavior from the dual-consistent algorithm is stable, which guarantees the precisions would be better with the refinement in this framework.

翻译：对偶一致性是发展面向目标网格自适应性的双重加权残差误差估计的重要问题，本文从理论和数值两方面对其进行了研究。基于牛顿-GMG求解器，详细讨论了求解稳态欧拉方程的对偶一致性。理论上，基于Petrov-Galerkin方法，深入研究了原问题与对偶问题，以及对偶一致性。研究发现，对偶一致性对于误差估计和感兴趣量的稳定收敛速率均具有重要意义。数值上，通过边界修正技术，可以确保一般构型问题的对偶一致性。数值实验清晰表明，在牛顿-GMG框架中关注对偶一致性的优势：与对偶不一致的实现相比，在计算感兴趣量时预计可节省一个数量级的网格单元。此外，对偶一致算法的收敛行为稳定，确保在该框架下随着网格细化精度将得到提升。