On the combined effect of boundary approximation and numerical integration on mixed finite element solution of 4th order elliptic problems with variable coefficients
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 33 (1999) no. 4, pp. 807-836.
@article{M2AN_1999__33_4_807_0,
author = {Bhattacharyya, Pulin K. and Nataraj, Neela},
title = {On the combined effect of boundary approximation and numerical integration on mixed finite element solution of 4th order elliptic problems with variable coefficients},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {807--836},
publisher = {EDP-Sciences},
volume = {33},
number = {4},
year = {1999},
zbl = {0942.65133},
mrnumber = {1726487},
language = {en},
url = {http://archive.numdam.org/item/M2AN_1999__33_4_807_0/}
}
Bhattacharyya, Pulin K.; Nataraj, Neela. On the combined effect of boundary approximation and numerical integration on mixed finite element solution of 4th order elliptic problems with variable coefficients. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 33 (1999) no. 4, pp. 807-836. http://archive.numdam.org/item/M2AN_1999__33_4_807_0/

[1] R.A. Adams, Sobolev Spaces. Academic Press, NewYork (1975). | MR 450957 | Zbl 0314.46030

[2] I. Babuska, Error Bounds for Finite Element Method. Numer. Math. 16 (1971) 322-333. | EuDML 132037 | MR 288971 | Zbl 0214.42001

[3] I. Babuska and J.E. Osborn, Eigenvalue Problems, in Handbook of Numerical Analysis, Vol. 2, Part 1, J.L. Lions and P.G. Ciarlet Eds., North-Holland, Amsterdam (1991) 641-787. | MR 1115240 | Zbl 0875.65087

[4] S. Balasundaram and P.K. Bhattacharyya, On Existence of Solution of the Dirichlet Problem of Fourth Order Partial Differential Equations with Variable Coefficients. Quart. Appl. Math. 39 (1983) 311-317. | MR 721421 | Zbl 0533.35024

[5] S. Balasundaram and P.K. Bhattacharyya, A Mixed Finite Element Method for Fourth Order Elliptic Equations with Variable Coefficients. Comput. Math. Appl 10 (1984) 245-256. | MR 757239 | Zbl 0553.65080

[6] S. Balasundaram and P.K. Bhattacharyya, A Mixed Finite Element Method for Fourth Order Elliptic Operators with Variable Coefficients, 4th Int. Symp. on Finite Element Methods in Flow Problems, Chuo University, Tokyo (1982), in Finite Element Analysis of Flow Problems, T. Kawai Ed., Tokyo University Press (1982) 995-1001. | Zbl 0508.76006

[7] S. Balasundaram and P.K. Bhattacharyya, Mixed Finite Element Method for Fourth Order Partial Differential Equations. ZAMM 66 (1986) 489-499. | MR 870849 | Zbl 0616.73064

[8] M. Bernadou, Méthodes d'éléments finis pour les problèmes de coques minces. Collection Recherches en Mathématiques Appliquées, Masson, Paris (1994).

[9] P.K. Bhattacharyya and N. Nataraj, Error Estimates for Isoparametric Mixed Finite Element Solution of 4th Order Elliptic Problems with Variable Coefficients (submitted). | Zbl 1006.65123

[10] P.K. Bhattacharyya, Mixed Finite Element Method for Fourth Order Elliptic Operators with Variable Coefficients, In Proc. of 6th Joint France-Italy- USSR Symp. on Numerical Solution of Nonlinear Problems, INRIA, France (1983) 127-144; Russian Translation, in Methods of Computational Mathematics and Mathematical Modelling, Computational Mathematics Section, Academy of Sciences, Moscow, USSR (1985) 76-99. | MR 802809 | Zbl 0671.73055

[11] P.K. Bhattacharyya, S. Gopalsamy and S. Balasundaram, On a Mixed Finite Element Method for Clamped Anisotropic Plate Bending Problems. Internat. J. Numer. Methods Engrg. 28 (1989) 1351-1370. | Zbl 0705.73227

[12] P.K. Bhattacharyya and S. Balasundaram, A Mixed Finite Element Method for Fourth Order Elliptic Problems with Variable Coefficients. J. Comput. Appl. Math. 22 (1988) 1-24. | MR 948883 | Zbl 0647.65075

[13] F. Brezzi, On the Existence, Uniqueness and Approximation of Saddle-Point Problems Arising from Lagrangian Multipliers. RAIRO Anal. Numér. 8 (1974) 129-151. | Numdam | MR 365287 | Zbl 0338.90047

[14] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag (1991). | MR 1115205 | Zbl 0788.73002

[15] F. Brezzi and P.A. Raviart, Mixed Finite Element Methods for 4th Order Elliptic Equations, in Topics in Nufnerical Analysis III, J. Miller Ed., Academic Press, New York (1978) 33-56. | MR 657975 | Zbl 0434.65085

[16] P.G. Ciarlet, Basic Error Estimates for Elliptic Problems, in Handbook of Numerical Analysis, Vol. 2, Part 1, J.L. Lions and P.G. Ciarlet Eds., North-Holland, Amsterdam (1991). | MR 1115237 | Zbl 0875.65086

[17] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). | MR 520174 | Zbl 0383.65058

[18] P.G. Ciarlet and P.A. Raviart, A Mixed Finite Element Method for the Biharmonic Equation, in Symp. on Mathematical Aspects of Finite Elements in Partial Differential Equations, C. de Boor Ed., Academic Press, New York (1974) 125-145. | MR 657977 | Zbl 0337.65058

[19] P.G. Ciarlet and P.A Raviart, The Combined Effect of Curved Boundaries and Numerical Integration in Isoparametric Finite Element Methods, in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, A.K. Aziz Ed., Academic Press, New York (1972) 409-474. | MR 421108 | Zbl 0262.65070

[20] S. Gopalsamy and P.K. Bhattacharyya, On Existence and Uniqueness of Solution of Boundary Value Problems of Fourth Order Elliptic Partial Differential Equations with Variable Coefficients. J. Math. Anal. Appl. 136 (1988) 589-608. | MR 972159 | Zbl 0673.35022

[21] P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985). | MR 775683 | Zbl 0695.35060

[22] K. Hellan, Analysis of Elastic Plates in Flexure by a Simplified Finite Element Method. Acta Polytech. Scand., Civil Engineering Series, Trondheim, 46 (1967). | Zbl 0237.73046

[23] L. Herrmann, Finite Element Bending Analysis for Plates. J. Eng. Mech. Div. ASCE 93, EM 5 (1967) 49-83.

[24] V.A. Kondratev, Boundary Value Problems for Elliptic Equations in Domains with Conical or Angular Points. Trudy Moskov. Mat. Obshch. 16 (1967) 7209-292. | MR 226187 | Zbl 0194.13405

[25] S.G. Leknitskii, Anisotropic Plates. Gordon and Breach Science Publishers, New York (1968).

[26] J.L. Lions, Problèmes aux Limites dans les Équations aux Dérivées Partielles. Les Presses de l'Université de Montréal, Montréal (1965). | MR 251372 | Zbl 0143.14003

[27] L. Mansfield, Approximation of the Boundary in the Finite Element solution of fourth order problems. SIAM J. Numer. Anal. 15 (1978) 568-579. | MR 471373 | Zbl 0391.65047

[28] T. Miyoshi, A finite element method for the solution of fourth order partial differential equations. Kumamoto J. Sci. (Math.) 9 (1973) 87-116. | MR 386298 | Zbl 0249.35007

[29] N. Nataraj, P.K. Bhattacharyya, S. Balasundaram and S. Gopalsamy, On a Mixed-Hybrid Finite Element Method for Anisotropic Plate Bending Problems. Internat. J. Numer. Methods Engrg. 39 (1996) 4063-4089. | MR 1420777 | Zbl 0882.73069

[30] J.T. Oden and G.F Carey, Finite Elements-Mathematical Aspects IV, The Texas Finite Element Series. Prentice Hall, Eaglewood Cliffs, New Jersey (1983). | MR 767806 | Zbl 0496.65055

[31] A.B. Ouaritini, Méthodes d'éléments finis mixtes pour des problèmes de coques minces. Ph.D. thesis, University of Pau et Pays de L'Adour, France (1984).

[32] P.A. Raviart and J.M. Thomas, Introduction à l'Analyse Numérique des Équations aux Dérivées Partielles. Mason, Paris (1983). | Zbl 0561.65069

[33] J.E Roberts and J.M. Thomas, Mixed and Hybrid Methods, in Handbook of Numerical Analysis, Vol. 2, Part 1, J.L Lions and P.G. Ciarlet Eds., North-Holland, Amsterdam (1991) 523-633. | MR 1115239 | Zbl 0875.65090

[34] R. Scholtz, A Mixed Method for 4th order Problems Using Linear Finite Elements. RAIRO Anal Numér. 12 (1978) 85-90. | Numdam | MR 483557 | Zbl 0382.65059

[35] R. Scott, Finite Element Techniques for Curved Boundaries. Ph.D. thesis, M.I.T. (1973).

[36] G. Strang and G.J. Fix, An Analysis of the Finite Element Method. Prentice-Hall, New York (1973). | MR 443377 | Zbl 0356.65096

[37] S. Timoshenko and S. Woinowsky-Kreiger, Theory of Plates and Shells. McGraw-Hill Book Company, New York (1959). | JFM 66.1049.02

[38] A. Zeníšek, Nonlinear Elliptic and Evolution Problems and their Finite Element Approximations. Academic Press, New York (1990). | MR 1086876 | Zbl 0731.65090

[39] M. Zlamal, Curved elements in the finite element method, I. SIAM J. Numer. Anal. 10 (1973) 229-240. | MR 395263 | Zbl 0285.65067

[40] M. Zlamal, Curved elements in the finite element method, II. SIAM J. Numer. Anal. 11 (1974) 347-362. | MR 343660 | Zbl 0277.65064