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Linear finite element superconvergence on simplicial meshes


Authors: Jie Chen, Desheng Wang and Qiang Du
Journal: Math. Comp. 83 (2014), 2161-2185
MSC (2010): Primary 65N30, 65N50
DOI: https://doi.org/10.1090/S0025-5718-2014-02810-X
Published electronically: February 12, 2014
MathSciNet review: 3223328
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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
Email: qdu@math.psu.edu

DOI: https://doi.org/10.1090/S0025-5718-2014-02810-X
Keywords: Superconvergence, finite element methods, simplicial meshes, edge patches, edge pair condition
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.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.