The self-equilibration of residuals and complementary a posteriori error estimates in the finite element method. (English) Zbl 0575.65100

Kelly, D. W.

Int. J. Numer. Methods Eng. 20, 1491-1506 (1984).

It is demonstrated that the residual in a compatible (displacement) finite element solution can be partitioned into local self-equilibrating systems on each element. An a posteriori error analysis is then based on a complementary approach and examples indicate that the guaranteed upper bound on the energy of the error is preserved.

MSC:

65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N15	Error bounds for boundary value problems involving PDEs
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation

Keywords:

Poisson equation; finite element method; a posteriori error analysis

References:

[1]	Babuška, Int. j. numer. methods eng. 12 pp 1597– (1978)
[2]	Babuška, Comput. Meths. Appl. Mech. Engng 17/18 pp 519– (1979)
[3]	’A-posteriori error analysis and adaptivity for the finite element method’, Ph.D. thesis, University College of Swansea, Wales, U.K. (1982).
[4]	Kelly, Int. j. numer. methods eng. 19 pp 1593– (1983)
[5]	and , Variational Methods in Theoretical Mechanics, Springer Verlag, Berlin and New York, 1976. · Zbl 0324.73001
[6]	’Direct variation techniques’, in Variational Methods in Engineering, Vol. 1 (Eds. and ), Southampton University Press, U.K., 1973.
[7]	, and , ’Hierarchical finite elements, error estimates and adaptivity’, in The Mathematics of Finite Elements and Applications IV (MAFELAP 1981) (Ed. ), Academic Press, London and New York, 1981.