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Mesh dependent stability and condition number estimates for finite element approximations of parabolic problems


Authors: Liyong Zhu and Qiang Du
Journal: Math. Comp. 83 (2014), 37-64
MSC (2010): Primary 65N30, 65F10
DOI: https://doi.org/10.1090/S0025-5718-2013-02703-2
Published electronically: May 2, 2013
MathSciNet review: 3120581
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Abstract: In this paper, we discuss the effects of spatial simplicial meshes on the stability and the conditioning of fully discrete approximations of a parabolic equation using a general finite element discretization in space with explicit or implicit marching in time. Based on the new mesh dependent bounds on extreme eigenvalues of general finite element systems defined for simplicial meshes, we derive a new time step size condition for the explicit time integration schemes presented, which provides more precise dependence not only on mesh size but also on mesh shape. For the implicit time integration schemes, some explicit mesh-dependent estimates of the spectral condition number of the resulting linear systems are also established. Our results provide guidance to the studies of numerical stability for parabolic problems when using spatially unstructured adaptive and/or possibly anisotropic meshes.


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Additional Information

Liyong Zhu
Affiliation: LMIB and School of Mathematics and Systems Sciences, Beihang University, 100191, Beijing, People’s Republic of China
Email: liyongzhu@buaa.edu.cn

Qiang Du
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
Email: qdu@math.psu.edu

DOI: https://doi.org/10.1090/S0025-5718-2013-02703-2
Keywords: Stable time step size, condition number, mesh quality, finite element method, unstructured mesh, parabolic problem
Received by editor(s): January 5, 2011
Received by editor(s) in revised form: November 16, 2011, and April 15, 2012
Published electronically: May 2, 2013
Additional Notes: The first author is supported in part by the National Natural Science Foundation of China (Nos.11001007, 91130019) and Research Fund for the Doctoral Program of Higher Education of China (No. 20101102120031) and ISTCP of China (No. 2010DFR00700).
The second author is supported in part by NSF DMS-1016073. Part of this work was completed during this author’s visit to the Beijing Computational Science Research Center, China.
Article copyright: © Copyright 2013 American Mathematical Society