An iterative finite element method for approximating the biharmonic equation

Author:
P. B. Monk

Journal:
Math. Comp. **51** (1988), 451-476

MSC:
Primary 65N15; Secondary 65N30

DOI:
https://doi.org/10.1090/S0025-5718-1988-0935080-0

MathSciNet review:
935080

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**[1]**Robert A. Adams,*Sobolev spaces*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR**0450957****[2]**Owe Axelsson,*Solution of linear systems of equations: iterative methods*, Sparse matrix techniques (Adv. Course, Technical Univ. Denmark, Copenhagen, 1976) Springer, Berlin, 1977, pp. 1–51. Lecture Notes in Math., Vol. 572. MR**0448834****[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 (Proc. Conf., Prague, 1962), Publ. House Czechoslovak Acad. Sci., Prague; Academic Press, New York, 1963, pp. 13–26. MR**0170133****[4]**Ivo Babuška and A. K. Aziz,*Survey lectures on the mathematical foundations of the finite element method*, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 1–359. With the collaboration of G. Fix and R. B. Kellogg. MR**0421106****[5]**J. J. Blair,*Higher order approximations to the boundary conditions for the finite element method*, Math. Comput.**30**(1976), no. 134, 250–262. MR**0398123**, https://doi.org/10.1090/S0025-5718-1976-0398123-3**[6]**James H. Bramble,*The Lagrange multiplier method for Dirichlet’s problem*, Math. Comp.**37**(1981), no. 155, 1–11. MR**616356**, https://doi.org/10.1090/S0025-5718-1981-0616356-7**[7]**James H. Bramble and Richard S. Falk,*Two mixed finite element methods for the simply supported plate problem*, RAIRO Anal. Numér.**17**(1983), no. 4, 337–384 (English, with French summary). MR**713765**, https://doi.org/10.1051/m2an/1983170403371**[8]**James H. Bramble and Ridgway Scott,*Simultaneous approximation in scales of Banach spaces*, Math. Comp.**32**(1978), no. 144, 947–954. MR**501990**, https://doi.org/10.1090/S0025-5718-1978-0501990-5**[9]**Philippe G. Ciarlet,*The finite element method for elliptic problems*, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR**0520174****[10]**P. G. Ciarlet and R. Glowinski,*Dual iterative techniques for solving a finite element approximation of the biharmonic equation*, Comput. Methods Appl. Mech. Engrg.**5**(1975), 277–295. MR**0373321**, https://doi.org/10.1016/0045-7825(75)90002-X**[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. 125-145.**[12]**Philip J. Davis and Philip Rabinowitz,*Methods of numerical integration*, 2nd ed., Computer Science and Applied Mathematics, Academic Press, Inc., Orlando, FL, 1984. MR**760629****[13]**Richard S. Falk,*Approximation of the biharmonic equation by a mixed finite element method*, SIAM J. Numer. Anal.**15**(1978), no. 3, 556–567. MR**0478665**, https://doi.org/10.1137/0715036**[14]**R. S. Falk and J. E. Osborn,*Error estimates for mixed methods*, RAIRO Anal. Numér.**14**(1980), no. 3, 249–277 (English, with French summary). MR**592753****[15]**R. Glowinski and O. Pironneau,*Numerical methods for the first biharmonic equation and the two-dimensional Stokes problem*, SIAM Rev.**21**(1979), no. 2, 167–212. MR**524511**, https://doi.org/10.1137/1021028**[16]**L. Herrmann, "Finite element bending analysis for plates,"*J. Eng. Mech. Div. A.S.C.E. EM*5, v. 93, 1967, pp. 49-83.**[17]**Claes Johnson,*On the convergence of a mixed finite-element method for plate bending problems*, Numer. Math.**21**(1973), 43–62. MR**0388807**, https://doi.org/10.1007/BF01436186**[18]**Tetsuhiko Miyoshi,*A finite element method for the solutions of fourth order partial differential equations*, Kumamoto J. Sci. (Math.)**9**(1972/73), 87–116. MR**0386298****[19]**Peter Monk,*A mixed finite element method for the biharmonic equation*, SIAM J. Numer. Anal.**24**(1987), no. 4, 737–749. MR**899701**, https://doi.org/10.1137/0724048**[20]**P. B. Monk,*Some Finite Element Methods for the Approximation of the Biharmonic Equation*. Ph.D. thesis, Rutgers University, 1983.**[21]**Martin Schechter,*On 𝐿^{𝑝} estimates and regularity. II*, Math. Scand.**13**(1963), 47–69. MR**0188616**, https://doi.org/10.7146/math.scand.a-10688**[22]**Ridgway Scott,*Interpolated boundary conditions in the finite element method*, SIAM J. Numer. Anal.**12**(1975), 404–427. MR**0386304**, https://doi.org/10.1137/0712032**[23]**A. H. Stroud & D. Secrest,*Gaussian Quadrature Formulae*, Prentice-Hall, Englewood Cliffs, N.J., 1973.

Retrieve articles in *Mathematics of Computation*
with MSC:
65N15,
65N30

Retrieve articles in all journals with MSC: 65N15, 65N30

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1988-0935080-0

Keywords:
Iterative method,
mixed method,
error estimate,
biharmonic equation

Article copyright:
© Copyright 1988
American Mathematical Society