Finite element methods for elliptic equations using nonconforming elements

Baker, Garth A.

doi:10.1090/S0025-5718-1977-0431742-5

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
Full-text PDF Free Access

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.

Ivo Babuška and Miloš Zlámal, Nonconforming elements in the finite element method with penalty, SIAM J. Numer. Anal. 10 (1973), 863–875. MR 345432, DOI https://doi.org/10.1137/0710071

G. BAKER, Projection Methods for Boundary Value Problems for Elliptic and Parabolic Equations with Discontinuous Coefficients, Ph. D. Thesis, Cornell Univ., 1973.

James H. Bramble, Todd Dupont, and Vidar Thomée, Projection methods for Dirichlet’s problem in approximating polygonal domains with boundary-value corrections, Math. Comp. 26 (1972), 869–879. MR 343657, DOI https://doi.org/10.1090/S0025-5718-1972-0343657-7
J. H. Bramble and S. R. Hilbert, Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation, SIAM J. Numer. Anal. 7 (1970), 112–124. MR 263214, DOI https://doi.org/10.1137/0707006
J. H. Bramble and A. H. Schatz, Least squares methods for $2m$th order elliptic boundary-value problems, Math. Comp. 25 (1971), 1–32. MR 295591, DOI https://doi.org/10.1090/S0025-5718-1971-0295591-8
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). MR 0247243
J. Nitsche, Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Sem. Univ. Hamburg 36 (1971), 9–15 (German). MR 341903, DOI https://doi.org/10.1007/BF02995904
Martin Schechter, On $L^{p}$ estimates and regularity. II, Math. Scand. 13 (1963), 47–69. MR 188616, DOI https://doi.org/10.7146/math.scand.a-10688
Gilbert Strang, The finite element method and approximation theory, Numerical Solution of Partial Differential Equations, II (SYNSPADE 1970) (Proc. Sympos., Univ. of Maryland, College Park, Md., 1970) Academic Press, New York, 1971, pp. 547–583. MR 0287723