×

Local error estimates of FEM for displacements and stresses in linear elasticity by solving local Neumann problems. (English) Zbl 1057.74042

Summary: We present two types of local error estimators for the primal finite element method (FEM) by duality arguments. They are first derived from the (explicit) residual error estimation method (REM) and then – as a new contribution – from the (implicit) posterior equilibrium method (PEM) using improved boundary tractions, gained by local post-processing with local Neumann problems, with applications in elastic problems. For the displacements, a local error estimator with an upper bound is derived, and also a local estimator for stresses. Furthermore – for better numerical efficiency – the residua are projected energy-invariant onto reference elements where the local Neumann problems have to be solved. Comparative examples between REM and PEM-type local estimators show superior effectivity indices for the latter one.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74B05 Classical linear elasticity
65N15 Error bounds for boundary value problems involving PDEs
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Babu?ka, Computer Methods in Applied Mechanics and Engineering 61 pp 1– (1987) · Zbl 0593.65064
[2] Babu?ka, Society for Industrial and Applied Mathematics 15 pp 736– (1978)
[3] Babu?ka, Computer Methods in Applied Mechanics and Engineering 140 pp 1– (1997) · Zbl 0896.73055
[4] Cirak, Computer Methods in Applied Mechanics and Engineering 156 pp 351– (1998) · Zbl 0947.74062
[5] Prudhomme, Computer Methods in Applied Mechanics and Engineering 176 pp 313– (1999) · Zbl 0945.65123
[6] A feed-back approach to error control in finite element methods: Application to linear elasticity. Preprint 96-42 (SFB 359), Universität Heidelberg, 1996; 1-24.
[7] Johnson, Computer Methods in Applied Mechanics and Engineering 101 pp 143– (1992) · Zbl 0778.73071
[8] Stein, Computer Methods in Applied Mechanics and Engineering 150 pp 327– (1997) · Zbl 0926.74127
[9] Theorie und Numerik dimensions- und modelladaptiver Finite-Element-Methoden von Flächentragwerken. Forschungs- und Seminarberichte aus dem Bereich der Mechanik der Universität Hannover. Institut für Baumechanik und Numerische Mechanik, Hannover, 1996.
[10] Stein, Computer Methods in Applied Mechanics and Engineering 176 pp 363– (1999) · Zbl 0954.74072
[11] Ainsworth, Computer Methods in Applied Mechanics and Engineering 142 pp 1– (1997) · Zbl 0895.76040
[12] Bufler, Ingenieur Archiv 39 pp 248– (1970) · Zbl 0197.21901
[13] Ladevèze, Society for Industrial and Applied Mathematics 20 pp 485– (1983)
[14] Brink, Computer Methods in Applied Mechanics and Engineering 161 pp 77– (1998) · Zbl 0943.74062
[15] Stein, Computer Methods in Applied Mechanics and Engineering 10 pp 175– (1977) · Zbl 0347.73058
[16] Bank, Mathematics of Computation 44 pp 283– (1985)
[17] Walhorn, Festschrift Erwin Stein, Forschungs- und Seminarberichte aus dem Bereich der Mechanik der Universität Hannover, Institut für Baumechanik und Numerische Mechanik 4 pp 279– (1998)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.