A Two-Level Additive Schwarz Preconditioner for Local C0 Discontinuous Galerkin Methods of Kirchhof Plates

doi:10.1007/s42967-019-0003-1

Communications on Applied Mathematics and Computation ›› 2019, Vol. 1 ›› Issue (2): 167-185.doi: 10.1007/s42967-019-0003-1

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A Two-Level Additive Schwarz Preconditioner for Local C⁰ Discontinuous Galerkin Methods of Kirchhof Plates

Jianguo Huang¹, Xuehai Huang²

1 School of Mathematical Sciences, and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, China;
2 School of Mathematics, Shanghai University of Finance and Economics, Shanghai 200433, China

收稿日期:2018-06-04 修回日期:2018-08-14 出版日期:2019-06-20 发布日期:2019-06-20
通讯作者: Jianguo Huang, Xuehai Huang E-mail:jghuang@sjtu.edu.cn;huang.xuehai@sufe.edu.cn
基金资助:
The authors thank the referees for their valuable comments leading to improvement of an early version of the paper. The frst author was supported under NSFC (Grant no. 11571237). The second author was supported under NSFC (Grant no. 11771338), and the Fundamental Research Funds for the Central Universities.

A Two-Level Additive Schwarz Preconditioner for Local C⁰ Discontinuous Galerkin Methods of Kirchhof Plates

Jianguo Huang¹, Xuehai Huang²

1 School of Mathematical Sciences, and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, China;
2 School of Mathematics, Shanghai University of Finance and Economics, Shanghai 200433, China

Received:2018-06-04 Revised:2018-08-14 Online:2019-06-20 Published:2019-06-20
Contact: Jianguo Huang, Xuehai Huang E-mail:jghuang@sjtu.edu.cn;huang.xuehai@sufe.edu.cn
Supported by:
The authors thank the referees for their valuable comments leading to improvement of an early version of the paper. The frst author was supported under NSFC (Grant no. 11571237). The second author was supported under NSFC (Grant no. 11771338), and the Fundamental Research Funds for the Central Universities.

摘要/Abstract

摘要： A two-level additive Schwarz preconditioner based on the overlapping domain decomposition approach is proposed for the local C⁰ discontinuous Galerkin (LCDG) method of Kirchhof plates. Then with the help of an intergrid transfer operator and its error estimates, it is proved that the condition number is bounded by O(1 + (H⁴/δ⁴)), where H is the diameter of the subdomains and δ measures the overlap among subdomains. And for some special cases of small overlap, the estimate can be improved as O(1 + (H³/δ³)). At last, some numerical results are reported to demonstrate the high efciency of the two-level additive Schwarz preconditioner.

关键词: Kirchhof plate, C⁰ discontinuous Galerkin, Two-level additive Schwarz preconditioner, Intergrid transfer operator

Abstract: A two-level additive Schwarz preconditioner based on the overlapping domain decomposition approach is proposed for the local C⁰ discontinuous Galerkin (LCDG) method of Kirchhof plates. Then with the help of an intergrid transfer operator and its error estimates, it is proved that the condition number is bounded by O(1 + (H⁴/δ⁴)), where H is the diameter of the subdomains and δ measures the overlap among subdomains. And for some special cases of small overlap, the estimate can be improved as O(1 + (H³/δ³)). At last, some numerical results are reported to demonstrate the high efciency of the two-level additive Schwarz preconditioner.

Key words: Kirchhof plate, C⁰ discontinuous Galerkin, Two-level additive Schwarz preconditioner, Intergrid transfer operator

中图分类号:

65N30
65N55

Jianguo Huang, Xuehai Huang. A Two-Level Additive Schwarz Preconditioner for Local C⁰ Discontinuous Galerkin Methods of Kirchhof Plates[J]. Communications on Applied Mathematics and Computation, 2019, 1(2): 167-185.

参考文献 1.

Brenner, S.C.:A two-level additive Schwarz preconditioner for macro-element approximations of the plate bending problem. Houst. J. Math. 21, 823-844(1995)2. Brenner, S.C.:A two-level additive Schwarz preconditioner for nonconforming plate elements. Numer. Math. 72, 419-447(1996)3. Brenner, S.C.:Two-level additive Schwarz preconditioners for nonconforming fnite element methods. Math. Comput. 65, 897-921(1996)4. Brenner, S.C., Scott, L.R.:The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, New York (2008)5. Brenner, S.C., Sung, L.:C0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains. J. Sci. Comput. 22(23), 83-118(2005)6. Brenner, S.C., Wang, K.:Two-level additive Schwarz preconditioners for C0interior penalty methods. Numer. Math. 102, 231-255(2005)7. Ciarlet, P.G.:The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)8. Cockburn, B., Shu, C.-W.:The local discontinuous Galerkin method for time-dependent convection-difusion systems. SIAM J. Numer. Anal. 35(6), 2440-2463(1998)9. Dryja, M., Widlund, O.B.:Some domain decomposition algorithms for elliptic problems. In:Iterative Methods for Large Linear Systems, pp. 273-291. Academic Press, Boston (1990)10. Dryja, M., Widlund, O.B.:Domain decomposition algorithms with small overlap. SIAM J. Sci. Comput. 15, 604-620(1994)11. Engel, G., Garikipati, K., Hughes, T.J.R., Larson, M.G., Mazzei, L., Taylor, R.L.:Continuous/discontinuous fnite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity. Comput. Methods Appl. Mech. Eng. 191, 3669-3750(2002)12. Feng, K., Shi, Z.:Mathematical Theory of Elastic Structures. Springer, Berlin (1995)13. Feng, X., Karakashian, O.A.:Two-level additive Schwarz methods for a discontinuous Galerkin approximation of second order elliptic problems. SIAM J. Numer. Anal. 39, 1343-1365(2001)14. Feng, X., Karakashian, O.A.:Two-level non-overlapping Schwarz preconditioners for a discontinuous Galerkin approximation of the biharmonic equation. J. Sci. Comput. 22(23), 289-314(2005)15. Huang, J., Huang, X., Han, W.:A new C0 discontinuous Galerkin method for Kirchhof plates. Comput. Methods Appl. Mech. Eng. 199, 1446-1454(2010)16. Lasser, C., Toselli, A.:An overlapping domain decomposition preconditioner for a class of discontinuous Galerkin approximations of advection-difusion problems. Math. Comput. 72, 1215-1238(2003)17. Nepomnyaschikh, S.V.:On the application of the bordering method to the mixed boundary value problem for elliptic equations and on mesh norms in W21∕2(S). Sov. J. Numer. Anal. Math. Model. 4, 493-506(1989)18. Reddy, J.N.:Theory and Analysis of Elastic Plates and Shells, 2nd edn. CRC Press, New York (2006)19. Wells, G.N., Dung, N.T.:A C0 discontinuous Galerkin formulation for Kirchhof plates. Comput. Methods Appl. Mech. Eng. 196, 3370-3380(2007)20. Zhang, X.:Two-level Schwarz methods for the biharmonic problem discretized conforming C0 elements. SIAM J. Numer. Anal. 33, 555-570(1996)

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A Two-Level Additive Schwarz Preconditioner for Local C⁰ Discontinuous Galerkin Methods of Kirchhof Plates

A Two-Level Additive Schwarz Preconditioner for Local C⁰ Discontinuous Galerkin Methods of Kirchhof Plates

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