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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Linear finite element superconvergence on simplicial meshes
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by Jie Chen, Desheng Wang and Qiang Du PDF
Math. Comp. 83 (2014), 2161-2185 Request permission

Abstract:

We study the linear finite element gradient superconvergence on special simplicial meshes which satisfy an edge pair condition. This special geometric condition implies that for most simplexes in the mesh, the lengths of each pair of opposite edges in each 3-face are assumed to differ only by $O(h^{1+\alpha })$ for some constant $\alpha >0$, with $h$ being the mesh parameter. To analyze the interplant gradient superconvergence, we present a local error expansion formula in general $n$ dimensional space which also motivates the condition on meshes. In the three dimensional space, we show that the gradient of the linear finite element solution $u_h$ is superconvergent to the gradient of the linear interpolatant $u_I$ with an order $O(h^{1+\rho })$ for $0<\rho \leq \alpha$. Numerical examples are presented to verify the theoretical findings. While we illustrate that tetrahedral meshes satisfying the edge pair condition can often be produced in three dimension, we also show that this may not be the case in higher dimensional spaces.
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Additional Information
  • Jie Chen
  • Affiliation: Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong
  • Email: majchen@ust.hk
  • Desheng Wang
  • Affiliation: Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371
  • Email: desheng@ntu.edu.sg
  • Qiang Du
  • Affiliation: Pennsylvania State University, University Park, Pennsylvania 16802
  • MR Author ID: 191080
  • Email: qdu@math.psu.edu
  • Received by editor(s): September 29, 2012
  • Received by editor(s) in revised form: December 24, 2012
  • Published electronically: February 12, 2014
  • Additional Notes: This work was supported by Singapore AcRF RG59/08 (M52110092) and Singapore NRF 2007 IDM-IDM002-010, and US NSF DMS-1318586.
  • © Copyright 2014 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 83 (2014), 2161-2185
  • MSC (2010): Primary 65N30, 65N50
  • DOI: https://doi.org/10.1090/S0025-5718-2014-02810-X
  • MathSciNet review: 3223328