Boundary layer resolution in hierarchical plate modelling. (English) Zbl 0821.73030

Summary: The dimensional reduction of an elliptic model boundary value problem on a thin, plate-like domain of thickness \(2d\) is analysed. A class of lower-dimensional models with variable model orders in the interior and near the boundary of the plate is investigated and it is shown that for a sufficiently large model order in a \( O (d| ln d |)\)- neighbourhood of the edge the hierarchical models compensate for the boundary layers of the exact solution – in the sense that their asymptotic rate of convergence as \(d \to 0\) is the same as the optimal one for compatible, layer-free solutions.


74K20 Plates
35B25 Singular perturbations in context of PDEs
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[1] and , ’Survey lectures on the mathematical foundation of the finite element method’, in: The mathematical foundations of the finite element method with applications to partial differential equations (ed.) pp. 3-343, Academic Press, New York, 1972. · doi:10.1016/B978-0-12-068650-6.50006-X
[2] Babuška, SIAM J. Num. Anal. (1994)
[3] Plates and Junctions in Elastic Multistructures, Masson, Paris; Springer, New York, 1990.
[4] Jensen, SIAM J. Numer. Anal. 29 pp 1294– (1992)
[5] Morgenstern, Arch. Rat. Mech. Anal. 4 pp 145– (1959)
[6] Les Méthodes Directes en Théorie des Equations Elliptiques, Masson, Paris, 1967.
[7] and , ’Energy estimates relating different linearly elastic models of a cylindrical shell, part I: the membrane-dominated case, part II: The bending-dominated case, part III: the soft membrane case; Reports A 286, A 293 and A 299’, Institute of Mathematics, Helsinki University of Technology, Espoo, Finland, 1991.
[8] ’Hierarchic Models of Elliptic Boundary Value Problems on Thin Domains–a-posteriori Error Estimation and Fourier Analysis’, Habilitation Thesis, Stuttgart University, 1994.
[9] and , ’Boundary layer approximation in hierarchical beam- and plate models’; Research Report 92-08, Department of Mathematics, University of Maryland, Baltimore County, Baltimore, MD 21228-5398, May 1992 (in press in J. Elasticity).
[10] Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970. · Zbl 0207.13501
[11] Vogelius, Math. Comp 37 pp 31– (1981)
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