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An MP-DWR method for h-adaptive finite element methods

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Abstract

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++ 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.

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Acknowledgements

The research of G. Hu was partially supported by National Natural Science Foundation of China (Grant Nos. 11922120 and 11871489), FDCT of Macao SAR (0082/2020/A2), MYRG of University of Macau (MYRG2020-00265-FST) and Guangdong-Hong Kong-Macao Joint Laboratory for Data-Driven Fluid Mechanics and Engineering Applications (2020B1212030001).

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science and Technology, University of Macau, Macao, SAR, China

    Chengyu Liu & Guanghui Hu

  2. Zhuhai UM Science & Technology Research Institute, Zhuhai, Guangdong, China

    Guanghui Hu

  3. Guangdong-Hong Kong-Macao Joint Laboratory for Data-Driven Fluid Mechanics and Engineering Applications, University of Macau, Macao SAR, China

    Guanghui Hu

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  1. Chengyu Liu
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Correspondence to Guanghui Hu.

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Liu, C., Hu, G. An MP-DWR method for h-adaptive finite element methods. Numer Algor 94, 1309–1329 (2023). https://doi.org/10.1007/s11075-023-01536-7

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  • DOI: https://doi.org/10.1007/s11075-023-01536-7

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