## New superconvergent structures developed from the finite volume element method in 1D

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Xiang Wang, Junliang Lv and Yonghai Li
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## Abstract:

New superconvergence structures are introduced by the finite volume element method (FVEM), which gives us the freedom to choose the superconvergent points of the derivative (for odd order schemes) and the superconvergent points of the function value (for even order schemes) for $k\geq 3$. The general orthogonal condition and the modified M-decomposition (MMD) technique are established to prove the superconvergence properties of the new structures. In addition, the relationships between the orthogonal condition and the convergence properties for the FVE schemes are carried out in Table 1. The above results are also valid to $k=1,2$. For these cases, the new superconvergence structures are the same with the Gauss-Lobatto structure, which yields to the FVE schemes with Gauss-points-dependent dual meshes. Numerical results are given to illustrate the theoretical results.## References

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## Additional Information

**Xiang Wang**- Affiliation: School of Mathematics, Jilin University, Changchun 130012, China
- MR Author ID: 936154
- ORCID: 0000-0003-1504-0921
- Email: wxjldx@jlu.edu.cn
**Junliang Lv**- Affiliation: School of Mathematics, Jilin University, Changchun 130012, China
- Email: lvjl@jlu.edu.cn
**Yonghai Li**- Affiliation: School of Mathematics, Jilin University, Changchun 130012, China
- MR Author ID: 363086
- Email: yonghai@jlu.edu.cn
- Received by editor(s): January 5, 2020
- Received by editor(s) in revised form: May 25, 2020, and July 29, 2020
- Published electronically: December 9, 2020
- Additional Notes: The first author was supported in part by the NSFC through grant 11701211, the China Postdoctoral Science Foundation through grant 2017M620106. The second author was supported in part by the Science Challenge Project through grant TZ2016002 and the Jilin Provincial Natural Science foundation through grant 20200201259JC. The third author was supported in part by the NSFC through grant 12071177 and the Joint Fund of the NSFC and the NASF through grant U1630249
- © Copyright 2020 American Mathematical Society
- Journal: Math. Comp.
**90**(2021), 1179-1205 - MSC (2020): Primary 65N12, 65N08, 65N30
- DOI: https://doi.org/10.1090/mcom/3587
- MathSciNet review: 4232221