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The $ W^1_p$ stability of the Ritz projection on graded meshes


Author: Hengguang Li
Journal: Math. Comp. 86 (2017), 49-74
MSC (2010): Primary 65N30, 65N12
DOI: https://doi.org/10.1090/mcom/3101
Published electronically: April 13, 2016
MathSciNet review: 3557793
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Abstract: Consider the Poisson equation on a convex polygonal domain and the finite element method of degree $ m\geq 1$ associated with a family of graded meshes for possible singular solutions. We prove the stability of the Ritz projection onto the finite element space in $ W^1_p$, $ 1<p\leq \infty $. Consequently, we obtain finite element error estimates in $ W^1_p$ for $ 1<p\leq \infty $ and in $ L^p$ for $ 1<p<\infty $. The key to the analysis is the use of the ``index engineering'' methodology in modified Kondrat$ '$ev weighted spaces. We also mention possible extensions and applications of these results.


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Additional Information

Hengguang Li
Affiliation: Department of Mathematics, Wayne State University, Detroit, Michigan 48202
Email: hli@math.wayne.edu

DOI: https://doi.org/10.1090/mcom/3101
Received by editor(s): June 5, 2014
Received by editor(s) in revised form: October 31, 2014, and June 18, 2015
Published electronically: April 13, 2016
Additional Notes: The author was supported in part by the NSF Grants DMS-1158839, DMS-1418853, and by the Wayne State University Grants Plus Program.
Article copyright: © Copyright 2016 American Mathematical Society

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