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：在基于有限元框架的对偶加权残量法中，伽辽金正交性导致无法在求解原始方程的同一空间中求解对偶方程。文献中已有两种常见方法构造对偶问题的新空间：网格细化方法（$h$方法）和逼近多项式阶数提升方法（$p$方法）。本文提出一种基于多精度技术的新方法——即新有限元空间的构造采用与原始方程相同的配置，仅改变计算精度。本文详细论证了该方法的可行性。数值实验表明，该方法可通过C++模板便捷实现，且在计算效率与存储开销上较$h$方法与$p$方法均有显著提升。值得关注的是，本方法的性能与文献中高阶插值方法（$i$方法）相当。两种方法的结合有望进一步提升对偶加权残量法的计算效率。