Babuška, Ivo; Miller, A. A feedback finite element method with a posteriori error estimation. I: The finite element method and some basic properties of the a posteriori error estimator. (English) Zbl 0593.65064 Comput. Methods Appl. Mech. Eng. 61, 1-40 (1987). This paper is the first in a series of two in which we discuss some theoretical and practical aspects of a feedback finite element method for solving systems of linear second-order elliptic partial differential equations (with particular interest in classical linear elasticity). In this first part we introduce some nonstandard finite element spaces, which though based on the usual square bilinear elements, permit local mesh refinement. The algebraic structure of these spaces and their approximation properties are analyzed. An ”equivalent estimator” for the \(H^ 1\) finite element error is developed. In the second paper we shall discuss the asymptotic properties of the estimator and computational experience. Cited in 2 ReviewsCited in 104 Documents MSC: 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 74S05 Finite element methods applied to problems in solid mechanics 35J25 Boundary value problems for second-order elliptic equations 65N15 Error bounds for boundary value problems involving PDEs Keywords:a posteriori error estimator; feedback finite element method; local mesh refinement PDF BibTeX XML Cite \textit{I. Babuška} and \textit{A. Miller}, Comput. Methods Appl. Mech. Eng. 61, 1--40 (1987; Zbl 0593.65064) Full Text: DOI References: [1] Babuška, I., The selfadaptive approach in the finite element method, (), 125-142 [2] Babuška, I.; Rheinboldt, W.C., A posteriori error estimates for finite element computations, SIAM J. numer. anal., 15, 736-754, (1978) · Zbl 0398.65069 [3] Babuška, I.; Rheinboldt, W.C., Reliable error estimation and mesh adaptation for the finite element method, (), 67-108 [4] Babuška, I.; Miller, A., A posteriori error estimates and adaptive techniques for finite element method, () [5] Mesztenyi, C.; Szymczak, W., Tech. note BN-991, () [6] Noor, A.K.; Babuška, I., Quality assessment and control of finite element solutions, (), (to appear). · Zbl 0608.73072 [7] Rheinboldt, W.C., Feedback systems and adaptivity for numerical computations, (), 3-19 [8] Babuška, I.; Miller, A.; Vogelius, M., Adaptive method and error estimation for elliptic problems of structural mechanics, (), 35-56 [9] Babuška, I.; Vogelius, M., Feedback and adaptive finite element solution of one-dimensional boundary value problems, Numer. math., 44, 75-103, (1984) · Zbl 0574.65098 [10] Babuška, I., Feedback, adaptivity and a-posteriori estimates in finite elements: aims, theory and experience, (), 3-25 [11] Babuška, I.; Yu, D., Asymptotically exact a-posteriori error estimator for biquadratic elements, (), (to appear). · Zbl 0619.73080 [12] Bank, R.E.; Weiser, A., Some a-posteriori error estimators for elliptic partial differential equations, Math. comput., 44, 283-301, (1985) · Zbl 0569.65079 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.