An iterative finite element method for approximating the biharmonic equation
Author:
P. B. Monk
Journal:
Math. Comp. 51 (1988), 451476
MSC:
Primary 65N15; Secondary 65N30
MathSciNet review:
935080
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Abstract: A mixed finite element method for the biharmonic model of the simply supported and clamped plate is analyzed and error estimates are obtained. We show that the discrete problem may be solved efficiently by using the conjugate gradient method and a sequence of Dirichlet problems for Poisson's equation.
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P. B. Monk, Some Finite Element Methods for the Approximation of the Biharmonic Equation. Ph.D. thesis, Rutgers University, 1983.
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Martin
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II, Math. Scand. 13 (1963), 47–69. MR 0188616
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Ridgway
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method, SIAM J. Numer. Anal. 12 (1975),
404–427. MR 0386304
(52 #7162)
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A. H. Stroud & D. Secrest, Gaussian Quadrature Formulae, PrenticeHall, Englewood Cliffs, N.J., 1973.
 [1]
 R. A. Adams, Sobolev Spaces, Pure and Appl. Math., Vol. 65, Academic Press, New York, 1975. MR 0450957 (56:9247)
 [2]
 O. Axelsson, Solution of Linear Systems of Equations: Iterative Methods in Sparse Matrix Techniques, Lecture Notes in Math., Vol. 572, SpringerVerlag, New York, 1977. MR 0448834 (56:7139)
 [3]
 I. Babuška, "The theory of small changes in the domain of existence in the theory of partial differential equations and its applications," Differential Equations and Their Applications, Academic Press, New York, 1963, pp. 1326. MR 0170133 (30:373)
 [4]
 I. Babuška & A. K. Aziz, "Survey lecture on the mathematical foundations of the finite element method," The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, (A. K. Aziz, ed.), Academic Press, New York, 1972, pp. 5359. MR 0421106 (54:9111)
 [5]
 J. J. Blair, "Higher order approximations to the boundary conditions for the finite element method," Math. Comp., v. 30, 1976, pp. 250262. MR 0398123 (53:1978)
 [6]
 J. H. Bramble, "The Lagrange multiplier method for Dirichlet's problem," Math. Comp., v. 37, 1981, pp. 111. MR 616356 (83h:65119)
 [7]
 J. H. Bramble & R. S. Falk, "Two mixed finite element methods for the simply supported plate problem," RAIRO Numér. Anal., v. 17, 1983, pp. 337384. MR 713765 (85a:65155)
 [8]
 J. H. Bramble & R. Scott, "Simultaneous approximation in scales of Banach spaces," Math. Comp., v. 32, 1978, pp. 947954. MR 501990 (80a:65222)
 [9]
 P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Studies in Math. and Its Applications, Vol. 4, NorthHolland, Amsterdam, 1978. MR 0520174 (58:25001)
 [10]
 P. G. Ciarlet & R. Glowinski, "Dual iterative techniques for solving a finite element approximation to the biharmonic equation," Comput. Methods Appl. Mech. Engrg., v. 5, 1975, pp. 277295. MR 0373321 (51:9521)
 [11]
 P. G. Ciarlet & A. Raviart, "A mixed finite element method for the biharmonic equation," Mathematical Aspects of Finite Elements in Partial Differential Equations, (C. de Boor, ed.), Academic Press, New York, 1974, pp. 125145.
 [12]
 P. J. Davis & P. Rabinowitz, Methods of Numerical Integration, 2nd ed., Academic Press, 1984. MR 760629 (86d:65004)
 [13]
 R. S. Falk, "Approximation of the biharmonic equation by a mixed finite element method," SIAM J. Numer. Anal., v. 15, 1978, pp. 556567. MR 0478665 (57:18142)
 [14]
 R. S. Falk & J. E. Osborn, "Error estimates for mixed methods," RAIRO Anal. Numér., v. 14, 1980, pp. 249277. MR 592753 (82j:65076)
 [15]
 R. Glowinski & O. Pironneau, "Numerical methods for the first biharmonic equation and for the twodimensional Stokes problem," SIAM Rev., v. 21, 1979, pp. 167212. MR 524511 (80e:65101)
 [16]
 L. Herrmann, "Finite element bending analysis for plates," J. Eng. Mech. Div. A.S.C.E. EM5, v. 93, 1967, pp. 4983.
 [17]
 C. Johnson, "On the convergence of a mixed finite element method for plate bending problems," Numer. Math., v. 21, 1973, pp. 4362. MR 0388807 (52:9641)
 [18]
 T. Miyoshi, "A finite element method for the solution of fourth order partial differential equations," Kumamoto J. Sci. (Math.), v. 9, 1973, pp. 87116. MR 0386298 (52:7156)
 [19]
 P. B. Monk, "A mixed finite element method for the biharmonic equation," SIAM J. Numer. Anal., v. 24, 1987, pp. 737749. MR 899701 (88j:65259)
 [20]
 P. B. Monk, Some Finite Element Methods for the Approximation of the Biharmonic Equation. Ph.D. thesis, Rutgers University, 1983.
 [21]
 M. Schechter, "On estimates and regularity. II," Math. Scand., v. 13, 1963, pp. 4769. MR 0188616 (32:6052)
 [22]
 R. Scott, "Interpolated boundary conditions in the finite element method," SIAM J. Numer. Anal., v. 12, 1975, pp. 404427. MR 0386304 (52:7162)
 [23]
 A. H. Stroud & D. Secrest, Gaussian Quadrature Formulae, PrenticeHall, Englewood Cliffs, N.J., 1973.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198809350800
PII:
S 00255718(1988)09350800
Keywords:
Iterative method,
mixed method,
error estimate,
biharmonic equation
Article copyright:
© Copyright 1988
American Mathematical Society
