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Journal of Scientific Computing

Erschienen in:

01.07.2014

A Partition of Unity Method with Penalty for Fourth Order Problems

verfasst von: Christopher B. Davis

Erschienen in: Journal of Scientific Computing | Ausgabe 1/2014

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Abstract

A partition of unity method for fourth order problems is proposed. As a model problem, we focus on the biharmonic problem with either clamped or simply supported boundary conditions when the domain is a bounded polygon. The algorithm is presented, error estimates are made, and numerical results are shown to verify the error estimates.

Vorheriger Artikel Domain Decomposition Preconditioners for Discontinuous Galerkin Methods for Elliptic Problems on Complicated Domains

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Argyris, J.H., Fried, I., Scharpf, D.W.: The TUBA family of plate elements for the matrix displacement method. Aeronaut. J. R. Aeronaut. Soc. 72, 701–709 (1968)

Babuska, I., Banerjee, U.: Stable generalized finite element method (SGFEM). Comput. Methods Appl. Mech. Eng. 201–204, 91–111 (2012)CrossRefMathSciNet

Babuska, I., Banerjee, U., Osborn, J.E.: Survey of meshless and generalized finite element methods: a unified approach. Acta Numerica 12, 1–125 (2003)CrossRefMATHMathSciNet

Babuska, I., Banerjee, U., Osborn, J.E.: Generalized finite element methods—main ideas, results and perspective. Int. J. Comput. Methods 1, 67–103 (2004)CrossRefMATH

Babuska, I., Banerjee, U., Osborn, J.E.: Superconvergence in the generalized finite element method. Numerische Mathematik 107(3), 353–395 (2007)CrossRefMATHMathSciNet

de Barcellos, C.S., Mendonca, P.D.R., Duarte, C.A.: A \(C^k\) continuous generalized finite element formulation applied to laminated Kirchhoff plate model. Comput. Mech. 44, 377–393 (2009)CrossRefMATHMathSciNet

Bogner, F.K., Fox, R.L., Schmit L.A.: The generation of interelement compatible stiffness and mass matrices by the use of interpolation formulas. In: Proceedings of the Conference Matrix Methods in Structural Mechanics, pp. 397–444 (1965)

Blum, H., Rannacher, R.: On the boundary value problem of the biharmonic operator on domains with angular corners. Math. Methods Appl. Sci. 2, 556–581 (1980)CrossRefMATHMathSciNet

Bramble, J.H., Hilbert, S.R.: Estimation of linear functionals on Sobolev spaces with applications to Fourier transforms and spline interpolation. SIAM J. Numer. Anal. 7, 113–124 (1970)CrossRefMathSciNet

10.

Brenner, S.C., Davis, C.B., Sung, L.-Y.: A generalized finite element method for the displacement obstacle problem of clamped Kirchhoff plates. Submitted (2012)

11.

Brenner, S.C., Neilan, M., Sung, L.-Y.: Isoparametric \(C^0\) interior penalty methods for plate bending problems on smooth domains. Calcolo. doi:10.1007/s10092-012-0057-1 (2012)

12.

Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, New York (2008)CrossRefMATH

13.

Brenner, S.C., Sung, L.-Y.: \(C^0\) interior penalty methods for fourth order elliptic boundary value problems on polygonal domains. J. Sci. Comput. 22/23, 83–118 (2005)CrossRefMathSciNet

14.

Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North Holland, Amsterdam (1978)MATH

15.

Dauge, M.: Elliptic Boundary Value Problems on Corner Domains, Lecture Notes in Mathematics, vol. 1341. Springer, Berlin (1988)

16.

Davis, C.B.: Meshless Boundary Particle Methods for Boundary Integral Equations and Meshfree Particle Methods for Plates. PhD thesis. University of North Carolina at Charlotte (2011)

17.

Georgoulis, E.H., Houston, P.: Discontinuous Galerkin methods for the biharmonic problem. IMA J. Numer. Anal. 29(3), 573–594 (2009)CrossRefMATHMathSciNet

18.

Griebel, M., Schweitzer, M.A.: A particle-partition of unity method for the solution of elliptic, parabolic, and hyperbolic PDEs. SIAM J. Sci. Comput 22(3), 853–890 (2000)CrossRefMATHMathSciNet

19.

Griebel, M., Schweitzer, M.A.: A particle-partition of unity method part VII: adaptivity. In: Meshfree Methods for Partial Differential Equations III, Lecture Notes on Computer Science and Engineering, vol. 26, pp. 121–148 (2006)

20.

Grisvard, P.: Elliptic Problems in Non Smooth Domains. Pitman, Boston (1985)

21.

Gudi, T.: Some nonstandard error analysis of discontinuous Galerkin methods for elliptic problems. Calcolo 47(4), 239–261 (2010)CrossRefMATHMathSciNet

22.

Lasry, J., Pommier, J., Renard, Y., Salaün, M.: Extended finite element methods for thin cracked plates with Kirchhoff-Love theory. Int. J. Numer. Meth. Eng. 84, 1115–1138 (2010)CrossRefMATH

23.

Lasry, J., Renard, Y., Salaün, M.: Stress intensity factors computation for bending plates with extended finite element method. Int. J. Numer. Meth. Eng. 91(9), 909–928 (2012)CrossRef

24.

Li, H.: A note on the conditioning of a class of generalized finite element methods. Appl. Numer. Math. 62, 754–766 (2012)CrossRefMATHMathSciNet

25.

Li, Z.: An overview of the immersed interface method and its applications. Taiwan. J. Math. 7, 1–49 (2003)MATH

26.

Mazzucato, A., Nistor, V., Qu, Q.: Quasi-optimal rates of convergence for the generalized finite element method in polygonal domains. IMA, preprint (2012)

27.

Melenk, J.M., Babuška, I.: The partition of unity finite element method: basic theory and applications. Comput. Methods Appl. Mech. Eng. 139, 289–314 (1996)CrossRefMATH

28.

Monk, P.: A mixed finite element method for the biharmonic equation. SIAM J. Numer. Anal. 24, 737–749 (1987)CrossRefMATHMathSciNet

29.

Morley, L.S.D.: The triangular equilibrium element in the solution of plate bending problems. Aero. Quart. 19, 129–169 (1968)

30.

Nitsche, J.A.: Über ein variationsprinzip zur lösung von Dirichlet-problemen bei verwendung von teilräumen, die keinen randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg 36, 9–15 (1971)CrossRefMATHMathSciNet

31.

Oh, H.-S., Davis, C., Jeong, J.W.: Meshfree particle methods for thin plates. Comput. Methods Appl. Mech. Eng. 209–212, 156–171 (2012)CrossRefMathSciNet

32.

Oh, H.-S., Jeong, J.W., Hong, W.-T.: The generalized product partition of unity for the meshless methods. J. Comput. Phys. 229, 1600–1620 (2010)CrossRefMATHMathSciNet

33.

Oh, H.-S., Kim, J.G., Hong, W.-T.: The piecewise polynomial partition of unity functions for the generalized finite element methods. Comput. Methods Appl. Mech. Eng. 197, 3702–3711 (2008)CrossRefMATHMathSciNet

34.

Peskin, C.S.: The immersed boundary method. Acta Numer. 11, 479–517 (2002)CrossRefMATHMathSciNet

35.

Schweitzer, M.A.: An algebraic treatment of essential boundary conditions in the particle-partition of unity method. SIAM J. Sci. Comput. 31(2), 1581–1602 (2009)CrossRefMATHMathSciNet

36.

Sun, J.: A new family of high regularity elements. Numer. Methods Partial Differ. Equ. 28, 1–16 (2012)CrossRefMATH

37.

Zhang, L., Gerstenberger, A., Wang, X., Liu, W.K.: Immersed finite element method. Comput. Methods Appl. Mech. Eng. 193(21), 2051–2067 (2004)CrossRefMATHMathSciNet

Titel: A Partition of Unity Method with Penalty for Fourth Order Problems
verfasst von: Christopher B. Davis
Publikationsdatum: 01.07.2014
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 1/2014
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-013-9795-8

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