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$ 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


Authors: Ruijian He, Xinlong Feng and Zhangxin Chen
Journal: Math. Comp. 87 (2018), 1659-1688
MSC (2010): Primary 35Q30, 65N30
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.


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Additional Information

Ruijian He
Affiliation: Department of Chemical and Petroleum Engineering, Schulich School of Engineering, University of Calgary, Calgary AB, Canada T2N 1N4
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
Email: zhachen@ucalgary.ca

DOI: https://doi.org/10.1090/mcom/3266
Keywords: 3D Poisson equation, difference finite element method, non-uniform mesh, $P_1$-conforming element, $H^1$-superconvergence
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.
Article copyright: © Copyright 2017 American Mathematical Society

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