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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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  • 1. D. Arnold, F. Brezzi and M. Fortin, A stable finite element for the Stokes equations, Calcolo, 12 (1984), 337-344. MR 799997 (86m:65136)
  • 2. O. Axelsson and V. Barker, Finite Element Solution of Boundary Value Problems, Academic Press, London, 1983; reprinted as Classics Appl. Math. 35, SIAM, Philadelphia, 2001. MR 1856818 (2002g:65001)
  • 3. R Bank and L. Scott, On the conditioning of finite element equations with highly refined meshes, SIAM J. Numer. Anal., 26(1989), 1383-1394. MR 1025094 (90m:65192)
  • 4. M. Batdor, L. Freitag and C. Ollivier-Gooch, Computational study of the effect of unstructured and mesh quality on solution efficiency, AIAA, 13th CFD Conf, 1997.
  • 5. M. Berzins, Mesh quality: a function of geometry, error estimates or both? Engineering with Computers, 15 (1999), 236-247.
  • 6. S. Brenner and L. Scott, The mathematical theory of finite element Methods, 2nd edition, Springer-Verlag, New York, 2002. MR 1894376 (2003a:65103)
  • 7. W. Cao, On the error of linear interpolation and orientation, aspect ratio, and internal angles of a triangle, SIAM J. Numer. Anal., 43 (2005), 19-40. MR 2177954 (2006k:65023)
  • 8. W. Dörfler, The conditioning of the stiffness matrix for certain elements approximating the incompressibility condition in fluid dynamics, Numer. Math., 58(1990), 203-214. MR 1069279 (91k:65142)
  • 9. Q. Du, Z. Huang and D. Wang, Mesh and Solver Coadaptation in finite element mehods for anisotropic problems, Numerical Methods for Partial Differential Equations, 21 (2005), 859-874. MR 2140812 (2006f:65126a)
  • 10. Q. Du, D. Wang and L. Zhu, On Mesh Geometry and Stiffness Matrix Conditioning for General Finite Element Spaces, SIAM J. Numer. Anal. 47(2009), 1421-1444. MR 2497335 (2010b:65252)
  • 11. A. Ern, J.L. Guermond, Evaluation of the condition number in linear systems arising in finite element approximations, ESAIM: M2AN, 40(2006), 29-48. MR 2223503 (2007b:65119)
  • 12. L. Freitag and C. Ollivier-Gooch, A cost/benefit analysis of simplicial mesh improvement techniques as measured by solution efficiency, Int. J. Comp. Geo. Appl., 10 (2000), 361-382. MR 1791193
  • 13. I. Fried, Bounds on the spectral and maximum norms of the finite element stiffness, flexibility and mass matrices, Int. J. Solids Structures, 9 (1973), 1013-1034. MR 0345400 (49:10136)
  • 14. I. Fried, Numerical solution of differential equations, Academic Press, New York, 1979. MR 526039 (80d:65001)
  • 15. I. Harari and T. Hughes, What are C and h?: Inequalities for the analysis and design of finite element methods, Computer Methods in Applied Mechanics and Engineering, 97 (1992), 157-192. MR 1167711 (93g:65122)
  • 16. N. Hu, X.-Z. Guo, I. Katz, Bounds for eigenvalues and condition numbers in the p-version of the finite element method, 67(1998), 1423-1450. MR 1484898 (99a:65149)
  • 17. W.Z. Huang and R. D. Russell, Adaptive Moving Mesh Methods, (Springer, New York). 2011. MR 2722625 (2012a:65243)
  • 18. Z.-C. Li, H.-T. Huang, Effective condtion number for the finite element method using local mesh refinements, Applied Numerical Mathematics, 59(2009), 1779-1795. MR 2532444 (2010g:65213)
  • 19. J. I. Lin, Bounds on eigenvalues of finite element systems, International Journal for numerical methods in engineering, 32(1991), 957-967. MR 1128904 (92g:80001)
  • 20. J. I. Lin, An element eigenvalue theorem and its application for stable time step sizes, Computer Methods in Applied Mechanics and Engineering, 73 (1989), 283-294. MR 1016643 (90k:73006)
  • 21. A. Ramage and A. Wathen, On preconditioning for finite element equations on irregular grids, SIAM J. Matrix Anal. Appl. 15 (1994), 909-921. MR 1282702 (95d:65095)
  • 22. J. Shewchuk, What is a good linear finite element? Interpolation, conditioning, anisotropy and quality measures, 2003, Technical report, CS, UC Berkeley.
  • 23. G. Strang and G. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ, 1973. MR 0443377 (56:1747)
  • 24. V. Thomee, Galerkin Finite Element Methods for Parabolic Problems, Springer, Berlin, 2nd edition, 2006. MR 2249024 (2007b:65003)
  • 25. A. Wathen, Realistic eigenvalue bounds for the Galerkin mass matrix, IMA J. Numer. Anal., 7 (1987), 449-457. MR 968517 (90a:65246)
  • 26. L. Zhu and Q. Du, Mesh dependent stability for finite element approximations of parabolic problems with mass lumping, Journal of Computational and Applied Math, 236(2011), 801-811. MR 2853505
  • 27. O. Zienkiewicz, R. Taylor and J. Zhu, The Finite Element Method, Its Basis and Fundamentals, Sixth edition, 2005, Elsevier, Oxford.

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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

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