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Some remarks on Zienkiewicz-Zhu estimator. (English) Zbl 0806.73069
Summary: The Zienkiewicz-Zhu estimator [O. C. Zienkiewicz and J. Z. Zhu, Int. J. Numer. Methods Eng. 24, 337-357 (1987; Zbl 0602.73063)] is analyzed for the piecewise linear finite element approximate solution of an elliptic problem. The estimator is proved to be equivalent to the error estimator for the Poisson equation with a homogeneous Dirichlet boundary condition for any triangular regular mesh. No assumptions are needed about the regularity of the solution (i.e., solutions with corner singularities are not excluded). The estimator is also proved to be asymptotically exact on subdomains where the solution is smooth when parallel meshes are used. Therefore, its behavior is similar to that of other well-known estimators.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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