In a dual weighted residual method based on the finite element framework, the Galerkin orthogonality is an issue that prevents solving the dual equation in the same space as the one for the primal equation. In the literature, there have been two popular approaches to constructing a new space for the dual problem, i.e., refining mesh grids ($h$-approach) and raising the order of approximate polynomials ($p$-approach). In this paper, a novel approach is proposed for the purpose based on the multiple-precision technique, i.e., the construction of the new finite element space is based on the same configuration as the one for the primal equation, except for the precision in calculations. The feasibility of such a new approach is discussed in detail in the paper. In numerical experiments, the proposed approach can be realized conveniently with C++ \textit{template}. Moreover, the new approach shows remarkable improvements in both efficiency and storage compared with the $h$-approach and the $p$-approach. It is worth mentioning that the performance of our approach is comparable with the one through a higher order interpolation ($i$-approach) in the literature. The combination of these two approaches is believed to further enhance the efficiency of the dual weighted residual method.

翻译：在基于有限要素框架的双重加权剩余方法中,Galerkin orthologicality是一个无法在与原始等式相同的空间内解决双重等式的问题,在文献中,为双重问题建造新空间有两种受欢迎的方法,即精炼网状网格($h$-约罗亚)和提高约合多元米亚值(p$-approach)的顺序。在本文中,为基于多重精确技术的目的提出了一种新颖的方法,即建造新的有限空间的配置与原始等式的配置相同,但计算精确性除外。在文件中详细讨论了这种新方法的可行性。在数字实验中,拟议的方法可以通过C++\textit{template}来方便地实现。此外,与美元-方略和美元-approach相比,新的方法在效率和储存方面都取得了显著的改进。值得一提的是,新有限空间的建造以与原始等式等同的配置为基础,但计算方法的精确性除外。在文件中详细讨论了这种新方法的可行性。在数字实验中,拟议的方法是通过两个更高的方法的组合。</s>