A conforming finite element method for overlapping and nonmatching grids

Authors:
Yunqing Huang and Jinchao Xu

Journal:
Math. Comp. **72** (2003), 1057-1066

MSC (2000):
Primary 65F10, 65N30

Published electronically:
November 18, 2002

MathSciNet review:
1972727

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we propose a finite element method for nonmatching overlapping grids based on the partition of unity. Both overlapping and nonoverlapping cases are considered. We prove that the new method admits an optimal convergence rate. The error bounds are in terms of local mesh sizes and they depend on neither the overlapping size of the subdomains nor the ratio of the mesh sizes from different subdomains. Our results are valid for multiple subdomains and any spatial dimensions.

**1.**Y. Achdou and Y. Maday.

The mortar element method with overlapping subdomains.

In*Proceedings of the 12th International Conference on Domain Decomposition Methods*. www.ddm.org. CMP**2001:12****2.**M. Aftosmis, J. Melton, and M. Berger.

Adaptation and surface modeling for cartesian mesh methods.

In*12th AIAA CFD. Conf*, volume AIAA Paper 95-1725, San Diego, CA, June 1995.**3.**J. M. Melenk and I. Babuška,*The partition of unity finite element method: basic theory and applications*, Comput. Methods Appl. Mech. Engrg.**139**(1996), no. 1-4, 289–314. MR**1426012**, 10.1016/S0045-7825(96)01087-0**4.**I. Babuška and J. M. Melenk,*The partition of unity method*, Internat. J. Numer. Methods Engrg.**40**(1997), no. 4, 727–758. MR**1429534**, 10.1002/(SICI)1097-0207(19970228)40:4<727::AID-NME86>3.3.CO;2-E**5.**Ivo Babuška and Zhimin Zhang,*The partition of unity method for the elastically supported beam*, Comput. Methods Appl. Mech. Engrg.**152**(1998), no. 1-2, 1–18. Symposium on Advances in Computational Mechanics, Vol. 5 (Austin, TX, 1997). MR**1602763**, 10.1016/S0045-7825(97)00231-4**6.**F. B. Belgacem.

The mortar finite element method with lagrange multipliers.*Numer. Math.*, 1998.**7.**F. B. Belgacem.

The mortar finite element method with lagrange multiplierss.*Numer. Math.*, pages 173-197, 1999.**8.**F. B. Belgacem and Y. Maday.

The mortar element method for three dimensional finite elements.

, 31:289-302, 1997.**9.**T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, and P. Krysl.

Meshless methods: An overview and recent developments.*Comp. Meth. Appl. Mech. Engrg.*, 139:3-48, 1996.**10.**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****11.**C. Bernardi, Y. Maday, and A. T. Patera,*A new nonconforming approach to domain decomposition: the mortar element method*, Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. XI (Paris, 1989–1991) Pitman Res. Notes Math. Ser., vol. 299, Longman Sci. Tech., Harlow, 1994, pp. 13–51. MR**1268898****12.**D. Braess and W. Dahmen,*Stability estimates of the mortar finite element method for 3-dimensional problems*, East-West J. Numer. Math.**6**(1998), no. 4, 249–263. MR**1667306****13.**Xiao-Chuan Cai, Maksymilian Dryja, and Marcus Sarkis,*Overlapping nonmatching grid mortar element methods for elliptic problems*, SIAM J. Numer. Anal.**36**(1999), no. 2, 581–606. MR**1675261**, 10.1137/S0036142997323582**14.**Victor Ryaben′kii,*Nonreflecting time-dependent boundary conditions on artificial boundaries of varying location and shape*, Proceedings of the Fourth International Conference on Spectral and High Order Methods (ICOSAHOM 1998) (Herzliya), 2000, pp. 481–492. MR**1772926**, 10.1016/S0168-9274(99)00116-6**15.**Yun Qing Huang and Yan Ping Chen,*Superconvergence and asymptotically exact a posteriori error estimates for finite elements on a 𝐾-mesh*, Math. Numer. Sinica**16**(1994), no. 3, 278–285 (Chinese, with English and Chinese summaries); English transl., Chinese J. Numer. Math. Appl.**16**(1994), no. 4, 66–74. MR**1392852****16.**G. Chesshire and W. Henshaw.

Composite overlapping meshes for the solution of partial differential equations.*J. Comp. Phys.*, 90:1-64, 1990.**17.**W. D. Henshaw D. L. Brown and D. J. Quinlan.

Overture: An object-oriented framework for solving partial differential equations on overlapping grids.

Technical report, UCRL-JC-132017, 1999.**18.**C. Armando Duarte and J. Tinsley Oden,*𝐻-𝑝 clouds—an ℎ-𝑝 meshless method*, Numer. Methods Partial Differential Equations**12**(1996), no. 6, 673–705. MR**1419770**, 10.1002/(SICI)1098-2426(199611)12:6<673::AID-NUM3>3.0.CO;2-P**19.**W. Henshaw.*Part I: The numerical solution of hyperbolic systems of conservation laws; Part II: Composite overlapping grid techniques*.

PhD thesis, Dept. Appl. Math., California Institute of Technology, Pasadena, CA, 1985.**20.**William D. Henshaw,*A fourth-order accurate method for the incompressible Navier-Stokes equations on overlapping grids*, J. Comput. Phys.**113**(1994), no. 1, 13–25. MR**1278187**, 10.1006/jcph.1994.1114**21.**Y. Huang and J. Xu.

Convergence of a generalized finite element method for elliptic problems with highly oscillating coefficients.

Manuscript.**22.**Yu. A. Kuznetsov,*Efficient iterative solvers for elliptic finite element problems on nonmatching grids*, Russian J. Numer. Anal. Math. Modelling**10**(1995), no. 3, 187–211. MR**1343473**, 10.1515/rnam.1995.10.3.187**23.**Y. Kuznetsov.

Overlapping domain decomposition with non-matching grids.

In P. Bjostad, M. Espedal, and D. Keyes, editors,*Proceedings of the 9th International Conference on Domain Decomposition*, pages 64-76. Domain Decomposition Press, 1998.**24.**Wing Kam Liu, Sukky Jun, Shaofan Li, Jonathan Adee, and Ted Belytschko,*Reproducing kernel particle methods for structural dynamics*, Internat. J. Numer. Methods Engrg.**38**(1995), no. 10, 1655–1679. MR**1331505**, 10.1002/nme.1620381005**25.**Göran Starius,*Composite mesh difference methods for elliptic boundary value problems*, Numer. Math.**28**(1977), no. 2, 243–258. MR**0461941****26.**Joseph L. Steger and John A. Benek,*On the use of composite grid schemes in computational aerodynamics*, Proceedings of the first world congress on computational mechanics (Austin, Tex., 1986), 1987, pp. 301–320. MR**912522**, 10.1016/0045-7825(87)90045-4**27.**P. Le Tallec, T. Sassi, and M. Vidrascu,*Three-dimensional domain decomposition methods with nonmatching grids and unstructured coarse solvers*, Domain decomposition methods in scientific and engineering computing (University Park, PA, 1993) Contemp. Math., vol. 180, Amer. Math. Soc., Providence, RI, 1994, pp. 61–74. MR**1312378**, 10.1090/conm/180/01957**28.**Barbara I. Wohlmuth,*Discretization methods and iterative solvers based on domain decomposition*, Lecture Notes in Computational Science and Engineering, vol. 17, Springer-Verlag, Berlin, 2001. MR**1820470****29.**J. Xu.*Theory of Multilevel Methods*.

PhD thesis, Cornell University, 1989.**30.**Jinchao Xu,*Iterative methods by space decomposition and subspace correction*, SIAM Rev.**34**(1992), no. 4, 581–613. MR**1193013**, 10.1137/1034116

Retrieve articles in *Mathematics of Computation*
with MSC (2000):
65F10,
65N30

Retrieve articles in all journals with MSC (2000): 65F10, 65N30

Additional Information

**Yunqing Huang**

Affiliation:
Institute for Computational and Applied Mathematics and Department of Mathematics, Xiangtan University, Peoples Republic of China, 411105

Email:
huangyq@mail.xtu.edu.cn

**Jinchao Xu**

Affiliation:
Center for Computational Mathematics and Applications Pennsylvania State University, University Park, Pennsylvania 16803

Email:
xu@math.psu.edu, http:////www.math.psu.edu/xu

DOI:
http://dx.doi.org/10.1090/S0025-5718-02-01459-X

Keywords:
Nonmatching grid,
partition of unity,
finite element,
overlapping,
domain decomposition,
mortar

Received by editor(s):
May 30, 2001

Received by editor(s) in revised form:
November 7, 2001

Published electronically:
November 18, 2002

Additional Notes:
The work was subsidized by the special funds for Major State Basic Research Projects through Xiangtan University, PRC, and partially supported by NSF DMS-0074299 through Pennsylvania State University and the Center for Computational Mathematics and Applications

Article copyright:
© Copyright 2002
American Mathematical Society