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Analysis of the efficiency of an a posteriori error estimator for linear triangular finite elements. (English) Zbl 0759.65069
This paper addresses the problem of determining upper and lower bounds for the effectivity index of the a posteriori estimate of the error in the finite element method. These bounds are given explicitly for a certain concrete estimator for linear elements and unstructured triangular meshes applied to the Laplace equation with mixed boundary conditions.
These bounds depend strongly on the geometry of the triangles and (relatively weakly) on the smoothness of the solution. An example shows that the bounds are not over pessimistic. Another application concerns the linear elasticity problem where larger sensitivity with respect to minimal angle is shown.
Detailed numerical experimentation is to appear in a forthcoming paper by the first author, L. Plank, and the third author [Finite Elem. Anal. Des. 11, No. 4, 285-306 (1992) (to appear)].
Reviewer: P.Burda (Praha)

65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74S05 Finite element methods applied to problems in solid mechanics
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
74B10 Linear elasticity with initial stresses
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