$H^1$-Superconvergence of a difference finite element method based on the $P_1-P_1$-conforming element on non-uniform meshes for the 3D Poisson equation
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- by Ruijian He, Xinlong Feng and Zhangxin Chen;
- Math. Comp. 87 (2018), 1659-1688
- DOI: https://doi.org/10.1090/mcom/3266
- Published electronically: October 26, 2017
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Abstract:
In this paper, a difference finite element (DFE) method is presented for the 3D Poisson equation on non-uniform meshes by using the $P_1-P_1$-conforming element. This new method consists of combining the finite difference discretization based on the $P_1$-element in the $z$-direction with the finite element discretization based on the $P_1$-element in the $(x,y)$-plane. First, under the regularity assumption of $u\in H^3(\Omega )\cap H^1_0(\Omega )$ and $\partial _{zz}f\in L^2((0, L_3);$ $H^{-1}(\omega ))$, the $H^1$-superconvergence of the discrete solution $u_\tau$ in the $z$-direction to the first-order interpolation function $I_\tau u$ is obtained, and the $H^1$-superconvergence of the second-order interpolation function $I^2_{2\tau } u_\tau$ in the $z$-direction to $u$ is then provided. Moreover, the $H^1$-superconvergence of the DFE solution $u_h$ to the $H^1$-projection $R_hu_\tau$ of $u_\tau$ is deduced and the $H^1$-superconvergence of the second-order interpolation function $I^2_{2\tau }I^2_{2h} u_h$ to $u$ in the $((x,y),z)$-space is also established. Finally, numerical tests are presented to show the $H^1$-superconvergence results of the DFE method for the 3D Poisson equation under the above regularity assumption.References
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Bibliographic Information
- Ruijian He
- Affiliation: Department of Chemical and Petroleum Engineering, Schulich School of Engineering, University of Calgary, Calgary AB, Canada T2N 1N4
- MR Author ID: 1190404
- Email: hejian010@gmail.com
- Xinlong Feng
- Affiliation: College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, People’s Republic of China
- Email: fxlmath@gmail.com, fxlmath@xju.edu.cn
- Zhangxin Chen
- Affiliation: Department of Chemical and Petroleum Engineering, Schulich School of Engineering, University of Calgary, Calgary AB, Canada T2N 1N4
- MR Author ID: 246747
- Email: zhachen@ucalgary.ca
- Received by editor(s): April 4, 2016
- Received by editor(s) in revised form: January 11, 2017, and February 11, 2017
- Published electronically: October 26, 2017
- Additional Notes: This research was made possible by contributions from NCET-13-0988, the NSF of China (No. 11671345 and No. 11362021), the NSF of Xinjiang Province (No. 2016D01C058), NSERC/AIEES/Foundation CMG IRC in Reservoir Simulation, AITF (iCore) Chair in Reservoir Modelling, and the Frank and Sarah Meyer Foundation CMG Collaboration Centre.
- © Copyright 2017 American Mathematical Society
- Journal: Math. Comp. 87 (2018), 1659-1688
- MSC (2010): Primary 35Q30, 65N30
- DOI: https://doi.org/10.1090/mcom/3266
- MathSciNet review: 3787388