Finite element methods for elliptic equations using nonconforming elements

Author:
Garth A. Baker

Journal:
Math. Comp. **31** (1977), 45-59

MSC:
Primary 65N30

DOI:
https://doi.org/10.1090/S0025-5718-1977-0431742-5

MathSciNet review:
0431742

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Abstract | References | Similar Articles | Additional Information

Abstract: A finite element method is developed for approximating the solution of the Dirichlet problem for the biharmonic operator, as a canonical example of a higher order elliptic boundary value problem.

The solution is approximated by special choices of classes of discontinuous functions, piecewise polynomial functions, by virtue of a special variational formulation of the boundary value problem. The approximating functions are not required to satisfy the prescribed boundary conditions.

Optimal error estimates are derived in Sobolev spaces.

**[1]**I. BABUŠKA & M. ZLÁMAL, "Nonconforming elements in the finite element method with penalty,"*SIAM J. Numer. Anal.*, v. 10, 1973, pp. 863-875. MR**49**# 10168. MR**0345432 (49:10168)****[2]**G. BAKER,*Projection Methods for Boundary Value Problems for Elliptic and Parabolic Equations with Discontinuous Coefficients*, Ph. D. Thesis, Cornell Univ., 1973.**[3]**J. H. BRAMBLE, T. DUPONT & V. THOMÉE, "Projection methods for Dirichlet's problem in approximating polygonal domains with boundary-value corrections,"*Math. Comp.*, v. 26, 1972, pp. 869-879. MR**49**# 8397. MR**0343657 (49:8397)****[4]**J. H. BRAMBLE & S. R. HILBERT, "Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation,"*SIAM J. Numer. Anal.*, v. 7, 1970, pp. 112-124. MR**41**#7819. MR**0263214 (41:7819)****[5]**J. H. BRAMBLE & A. H. SCHATZ, "Least squares methods for 2*m*th order elliptic boundary-value problems,"*Math. Comp.*, v. 25, 1971, pp. 1-32. MR**45**# 4657. MR**0295591 (45:4657)****[6]**J. L. LIONS & E. MAGENES,*Problèmes aux Limites Non Homogènes et Applications*, Vols. 1, 2, Dunod, Paris, 1968. MR**40**# 512, # 513. MR**0247243 (40:512)****[7]**J. A. NITSCHE, "Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind,"*Abh. Math. Sem. Univ. Hamburg*, v. 36, 1971, pp. 9-15. MR**49**# 6649. MR**0341903 (49:6649)****[8]**M. SCHECHTER, "On estimates and regularity. II,"*Math. Scand.*, v. 13, 1963, pp. 47-69. MR**32**# 6052. MR**0188616 (32:6052)****[9]**G. STRANG, "The finite element method and approximation theory," in*Numerical Solution of Partial Differential Equations*, II (*SYNSPADE*1970), (Proc. Sympos., Univ. of Maryland, 1970), edited by B. E. Hubbard, Academic Press, New York, 1971, pp. 547-583. MR**44**# 4926. MR**0287723 (44:4926)**

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DOI:
https://doi.org/10.1090/S0025-5718-1977-0431742-5

Article copyright:
© Copyright 1977
American Mathematical Society