Uniform $hp$ convergence results for the mortar finite element method
HTML articles powered by AMS MathViewer
- by Padmanabhan Seshaiyer and Manil Suri PDF
- Math. Comp. 69 (2000), 521-546 Request permission
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.References
- 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.
- I. Babuška, A. Craig, J. Mandel, and J. Pitkäranta, Efficient preconditioning for the $p$-version finite element method in two dimensions, SIAM J. Numer. Anal. 28 (1991), no. 3, 624–661. MR 1098410, DOI 10.1137/0728034
- I. Babuška and Manil Suri, The optimal convergence rate of the $p$-version of the finite element method, SIAM J. Numer. Anal. 24 (1987), no. 4, 750–776. MR 899702, DOI 10.1137/0724049
- I. Babuška and Manil Suri, The $h$-$p$ version of the finite element method with quasi-uniform meshes, RAIRO Modél. Math. Anal. Numér. 21 (1987), no. 2, 199–238 (English, with French summary). MR 896241, DOI 10.1051/m2an/1987210201991
- Ivo Babuška and Manil Suri, The $p$ and $h$-$p$ versions of the finite element method, basic principles and properties, SIAM Rev. 36 (1994), no. 4, 578–632. MR 1306924, DOI 10.1137/1036141
- V. I. Shmyrëv, A class of linear homogeneous complementarity problems in $\textbf {R}^n_+$, Optimizatsiya 45(62) (1989), 54–65 (Russian). MR 1090928
- F. Ben Belgacem. The mortar finite element method with Lagrange multipliers. Numer. Math., to appear, 1998
- C. Bernardi, Y. Maday, and A. T. Patera, Domain decomposition by the mortar element method, Asymptotic and numerical methods for partial differential equations with critical parameters (Beaune, 1992) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 384, Kluwer Acad. Publ., Dordrecht, 1993, pp. 269–286. MR 1222428
- F. Brezzi and L. D. Marini, Macro hybrid elements and domain decomposition methods, Optimisation et contrôle (Sophia-Antipolis, 1992) Cépaduès, Toulouse, 1993, pp. 89–96. MR 1284966
- Mario A. Casarin and Olof B. Widlund, A hierarchical preconditioner for the mortar finite element method, Electron. Trans. Numer. Anal. 4 (1996), no. June, 75–88. MR 1401446
- Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
- M. Crouzeix and V. Thomée, The stability in $L_p$ and $W^1_p$ of the $L_2$-projection onto finite element function spaces, Math. Comp. 48 (1987), no. 178, 521–532. MR 878688, DOI 10.1090/S0025-5718-1987-0878688-2
- Monique Dauge, Elliptic boundary value problems on corner domains, Lecture Notes in Mathematics, vol. 1341, Springer-Verlag, Berlin, 1988. Smoothness and asymptotics of solutions. MR 961439, DOI 10.1007/BFb0086682
- Milo R. Dorr, On the discretization of interdomain coupling in elliptic boundary-value problems, Domain decomposition methods (Los Angeles, CA, 1988) SIAM, Philadelphia, PA, 1989, pp. 17–37. MR 992001
- S. Minakshi Sundaram, On non-linear partial differential equations of the parabolic type, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 479–494. MR 0000088
- B. Guo and I. Babuška. The $hp$ version of the finite element method. Comput. Mech., 1:21–41 (Part I) 203-220 (Part II), 1986.
- 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
- P.-A. Raviart and J. M. Thomas, Primal hybrid finite element methods for $2$nd order elliptic equations, Math. Comp. 31 (1977), no. 138, 391–413. MR 431752, DOI 10.1090/S0025-5718-1977-0431752-8
- L. Ridgway Scott and Shangyou Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp. 54 (1990), no. 190, 483–493. MR 1011446, DOI 10.1090/S0025-5718-1990-1011446-7
- P. Seshaiyer. Non-Conforming $hp$ finite element methods. Ph.D. Dissertation, University of Maryland Baltimore County, 1998.
- P. Seshaiyer and M. Suri. Convergence results for non-conforming $hp$ methods: The mortar finite element method. AMS, Contemporary Mathematics, 218:467-473, 1998.
- E. P. Stephan and M. Suri, The $h$-$p$ version of the boundary element method on polygonal domains with quasiuniform meshes, RAIRO Modél. Math. Anal. Numér. 25 (1991), no. 6, 783–807 (English, with French summary). MR 1135993, DOI 10.1051/m2an/1991250607831
- Howard Swann, On the use of Lagrange multipliers in domain decomposition for solving elliptic problems, Math. Comp. 60 (1993), no. 201, 49–78. MR 1149294, DOI 10.1090/S0025-5718-1993-1149294-2
- Dunham Jackson, A class of orthogonal functions on plane curves, Ann. of Math. (2) 40 (1939), 521–532. MR 80, DOI 10.2307/1968936
- 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.
- J. Xu and J. Zou. Non-overlapping domain decomposition methods. Submitted to SIAM Review.
Additional Information
- Padmanabhan Seshaiyer
- Affiliation: Department of Biomedical Engineering, Texas A&M University, College Station, TX 77843-3120
- MR Author ID: 637907
- Email: padhu@terminator.tamu.edu
- Manil Suri
- Affiliation: Department of Mathematics and Statistics, University of Maryland Baltimore County, Baltimore, MD 21250
- Email: suri@math.umbc.edu
- 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.
- © Copyright 2000 American Mathematical Society
- Journal: Math. Comp. 69 (2000), 521-546
- MSC (1991): Primary 65N30, 65N15
- DOI: https://doi.org/10.1090/S0025-5718-99-01083-2
- MathSciNet review: 1642762