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Uniform $hp$ convergence results
for the mortar finite element method

Authors: Padmanabhan Seshaiyer and Manil Suri
Journal: Math. Comp. 69 (2000), 521-546
MSC (1991): Primary 65N30, 65N15
Published electronically: February 26, 1999
MathSciNet review: 1642762
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Abstract | References | Similar Articles | Additional Information

Abstract: The mortar finite element is an example of a non-conforming method which can be used to decompose and re-compose a domain into subdomains without requiring compatibility between the meshes on the separate components. We obtain stability and convergence results for this method that are uniform in terms of both the degree and the mesh used, without assuming quasiuniformity for the meshes. Our results establish that the method is optimal when non-quasiuniform $h$ or $hp$ methods are used. Such methods are essential in practice for good rates of convergence when the interface passes through a corner of the domain. We also give an error estimate for when the $p$ version is used. Numerical results for $h,p$ and $hp$ mortar FEMs show that these methods behave as well as conforming FEMs. An $hp$ extension theorem is also proved.

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  • 1. M. A. Aminpour, S. L. McClearly, J. B. Ransom. A global/local analysis method for treating details in structural design. Proceedings of the Third NASA Advanced Composites Technology Conference, NASA CP-3178, Vol 1, Part 2, 967-986, 1992.
  • 2. I. Babu\v{s}ka, A. Craig, J. Mandel, and J. Pitkäranta. Efficient preconditionings for the $p$-version of the finite element method in two dimensions. SIAM J. Numer. Anal., 28:624-661, 1991. MR 92a:65282
  • 3. I. Babu\v{s}ka and M. Suri. The optimal convergence rate of the $p$-version of the finite element method. SIAM J. Numer. Anal., 24:750-776, 1987. MR 88k:65102
  • 4. I. Babu\v{s}ka and M. Suri. The $h$-$p$ version of the finite element method with quasiuniform meshes. RAIRO Math. Modeling Numer. Anal., 21:199-238, 1987. MR 88d:65154
  • 5. I. Babu\v{s}ka and M. Suri. The $p$ and $h$-$p$ versions of the finite element method: basic principles and properties. SIAM Review, 36:578-632, 1994. MR 96d:65184
  • 6. I. Babu\v{s}ka, B. Q. Guo and E. P. Stephan. On the exponential convergence of the $h$-$p$ version for boundary element Galerkin methods on polygons. Math. Methods Appl. Sci. 12 (1990), 413-427. MR 91m:65174
  • 7. F. Ben Belgacem. The mortar finite element method with Lagrange multipliers. Numer. Math., to appear, 1998
  • 8. C. Bernardi, Y. Maday, A. T. Patera. Domain decomposition by the mortar element method, in Asymptotic and Numerical Methods for PDEs with Critical Parameters. H. G. Kaper and M. Garbey (eds.), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 384, Kluwer, 269-286, 1993. MR 94c:65151
  • 9. F. Brezzi and L. D. Marini. Macro hybrid elements and domain decomposition methods. Proc. Colloque en l'honneur du 60eme anniversaire de Jean Cea, Sophia-Antipolis, 1992. Cépaduès Toulouse, 89-96, 1993. MR 95c:65208
  • 10. M. Casarin and O. B. Widlund A hierarchical preconditioner for the mortar finite element method. ETNA, Electron. Trans. Numer. Anal., 4:75-88, 1996. MR 97g:65088
  • 11. P. G. Ciarlet. The Finite Element Method for Elliptic Problems. North Holland, Amsterdam, 1978. MR 58:25001
  • 12. M. Crouzeix and V. Thomée. The stability in $L_p$ and $W_p^1$ of the $L_2$-projection onto finite element function spaces. Math. Comp., 48:521-532, 1987. MR 88f:41016
  • 13. M. Dauge. Elliptic Boundary Value Problems on Corner Domains. Lecture Notes in Math., 1341, Springer, New York, 1988. MR 91a:35078
  • 14. M. Dorr. On the discretization of inter-domain coupling in elliptic boundary-value problems via the $p$ version of the finite element method in Domain Decomposition Methods (T. F. Chan, R. Glowinski, J. Periaux, O. B. Widlund, editors). SIAM, 17-37, 1989. MR 90d:65191
  • 15. W. Gui and I. Babu\v{s}ka. The $h,p$ and $h$-$p$ versions of the finite element method in one dimension. Numer. Math., 3:577-657, 1986. MR 88b:65130
  • 16. B. Guo and I. Babu\v{s}ka. The $hp$ version of the finite element method. Comput. Mech., 1:21-41 (Part I) 203-220 (Part II), 1986.
  • 17. N. Hu, X. Guo, and I. N. Katz. Lower and upper bounds for eigenvalues and condition numbers in the $p$ version of FEM. SIAM J. Numer. Anal., to appear
  • 18. P. A. Raviart and J. M. Thomas. Primal hybrid finite element methods for 2nd order elliptic equations. Math. Comp., 31:391-396, 1977. MR 55:4747
  • 19. L. Scott and S. Zhang. Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp., 54:483-493, 1990. MR 90j:65021
  • 20. P. Seshaiyer. Non-Conforming $hp$ finite element methods. Ph.D. Dissertation, University of Maryland Baltimore County, 1998.
  • 21. P. Seshaiyer and M. Suri. Convergence results for non-conforming $hp$ methods: The mortar finite element method. AMS, Contemporary Mathematics, 218:467-473, 1998.
  • 22. E. P. Stephan and M. Suri. The $h$-$p$ version of the boundary element method for polygonal domains with quasiuniform meshes. RAIRO Math. Mod. Numer. Anal., 25:783-807, 1991. MR 92m:65154
  • 23. H. Swann. On the use of Lagrange multipliers in domain decomposition for solving elliptic problems. Math. Comp., 60:49-78, 1993. MR 93d:65101
  • 24. H. Triebel Interpolation Theory, Function Spaces, Differential Operators. North Holland, Amsterdam/New York/Oxford, 1978. MR 80i:46032
  • 25. O. B. Widlund. An extension theorem for finite element spaces with three applications, in Numerical Techniques in Continuum Mechanics. Proceedings of the GAMM seminar, W. Hackbush and K. Witsch eds., Kiel 1986.
  • 26. J. Xu and J. Zou. Non-overlapping domain decomposition methods. Submitted to SIAM Review.

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Additional Information

Padmanabhan Seshaiyer
Affiliation: Department of Biomedical Engineering, Texas A&M University, College Station, TX 77843-3120

Manil Suri
Affiliation: Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, MD 21250

Keywords: $p$ version, $hp$ version, mortar elements, finite elements, non-conforming
Received by editor(s): August 4, 1997
Received by editor(s) in revised form: April 7, 1998
Published electronically: February 26, 1999
Additional Notes: This work was supported in part by the Air Force Office of Scientific Research, Air Force Systems Command, USAF under Grant F49620-95-I-0230, and by the National Science Foundation under Grant DMS-9706594.
Article copyright: © Copyright 2000 American Mathematical Society

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