论文标题
MP-DWR方法，用于$ h $ - 适应有限元方法
An MP-DWR method for $h$-adaptive finite element methods
论文作者
论文摘要
在基于有限元框架的双重加权残差方法中，Galerkin正交性是一个问题，可以防止在与原始方程相同的空间中求解双方方程。在文献中，已经有两种流行的方法来构建双重问题的新空间，即精炼网格（$ h $ - 呼吸）并提高了近似多项式（$ p $ - apprach）的顺序。在本文中，提出了一种新的方法，以基于多精确技术的目的，即，新有限元空间的构建基于与原始方程相同的配置，除了计算的精度。本文详细讨论了这种新方法的可行性。在数值实验中，可以使用C ++ \ textIt {Template}方便地实现所提出的方法。此外，与$ h $ appraching和$ p $ appraching相比，新方法在效率和存储方面都显示出显着提高。值得一提的是，我们的方法的性能与文献中的高阶插值（$ i $ apprace）相媲美。人们认为，这两种方法的组合可以进一步提高双重加权残差方法的效率。
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.
