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
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.
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 |
References:
| [1] | Babuška, I., The selfadaptive approach in the finite element method, (Whiteman, J. R., The Mathematics of Finite Elements and Application II (1976), Academic Press: Academic Press New York), 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, (Oden, J. T., Computations Methods in Nonlinear Mechanics (1980), North-Holland: North-Holland Amsterdam), 67-108 · Zbl 0451.65078 |
| [4] | Babuška, I.; Miller, A., A posteriori error estimates and adaptive techniques for finite element method, (Tech. Note BN-968 (1981), Institute for Physical Science and Technology, University of Maryland: Institute for Physical Science and Technology, University of Maryland College Park, MD) |
| [5] | Mesztenyi, C.; Szymczak, W., Tech. Note BN-991, (FEARS user’s manual for Univac 1100 (1982), Institute for Physical Science and Technology, University of Maryland: Institute for Physical Science and Technology, University of Maryland College Park, MD) |
| [6] | Noor, A. K.; Babuška, I., Quality assessment and control of finite element solutions, (Finite Elements in Analysis and Design (1987)), (to appear). · Zbl 0608.73072 |
| [7] | Rheinboldt, W. C., Feedback systems and adaptivity for numerical computations, (Babuška, I.; Chandra, J.; Flaherty, J. E., Adaptive Computational Methods for Partial Differential Equations (1983), SIAM: SIAM Philadelphia, PA), 3-19 · Zbl 0584.65040 |
| [8] | Babuška, I.; Miller, A.; Vogelius, M., Adaptive method and error estimation for elliptic problems of structural mechanics, (Babuška, I.; Chandra, J.; Flaherty, J. E., Adaptive Computational Method for Partial Differential Equations (1983), SIAM: SIAM Philadelphia, PA), 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, (Babuška, I.; Zienkiewicz, O. C.; Gago, J.; de Arantes Oliveira, E. R., Accuracy Estimates and Adaptivity for Finite Elements (1986), Wiley: Wiley New York), 3-25 |
| [11] | Babuška, I.; Yu, D., Asymptotically exact a-posteriori error estimator for biquadratic elements, (Finite Elements in Analysis and Design (1987)), (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 |
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