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Superconvergence analysis of the linear finite element method and a gradient recovery postprocessing on anisotropic meshes


Author: Weiming Cao
Journal: Math. Comp. 84 (2015), 89-117
MSC (2010): Primary 65N30, 65N15, 65N50
DOI: https://doi.org/10.1090/S0025-5718-2014-02846-9
Published electronically: May 28, 2014
MathSciNet review: 3266954
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Abstract: For the linear finite element method based on general unstructured anisotropic meshes in two dimensions, we establish the superconvergence in energy norm of the finite element solution to the interpolation of the exact solution for elliptic problems. We also prove the superconvergence of the postprocessing process based on the global $ L^2$-projection of the gradient of the finite element solution. Our basic assumptions are: (i) the mesh is quasi-uniform under a Riemannian metric and (ii) each adjacent element pair forms an approximate (anisotropic) parallelogram. The analysis follows the same methodology developed by Bank and Xu in 2003 for the case of quasi-uniform meshes, and the results can be considered as an extension of their conclusion to the adaptive anisotropic meshes. Numerical examples involving both internal and boundary layers are presented in support of the theoretical analysis.


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Additional Information

Weiming Cao
Affiliation: Department of Mathematics, University of Texas at San Antonio, San Antonio, Texas 78249
Email: weiming.cao@utsa.edu

DOI: https://doi.org/10.1090/S0025-5718-2014-02846-9
Keywords: Superconvergence, anisotropic mesh, recovery type error estimates, post processing, linear finite element
Received by editor(s): June 30, 2012
Received by editor(s) in revised form: April 29, 2013
Published electronically: May 28, 2014
Additional Notes: This work was supported in part by NSF grant DMS-0811232.
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.