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Asymptotic Estimates of Hierarchical Modeling

Douglas N. Arnold

Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN 55455, USA

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and

Alexandre L. Madureira

Laboratório Nacional de Computação Científica, Petrópolis, RJ 25651-070, Brazil

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https://doi.org/10.1142/S0218202503002933Cited by:7 (Source: Crossref)

Abstract

In this paper we propose a way to analyze certain classes of dimension reduction models for elliptic problems in thin domains. We develop asymptotic expansions for the exact and model solutions, having the thickness as small parameter. The modeling error is then estimated by comparing the respective expansions, and the upper bounds obtained make clear the influence of the order of the model and the thickness on the convergence rates. The techniques developed here allows for estimates in several norms and semi-norms, and also interior estimates (which disregards boundary layers).

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References

D. N. Arnold and R. S. Falk, SIAM J. Math. Anal. 27, 468 (1996), DOI: 10.1137/S0036141093245276. Google Scholar
I. Babuška and M. Suri, SIAM J. Numer. Anal. 24, 750 (1987), DOI: 10.1137/0724049. Crossref, Web of Science, Google Scholar
I. Babuška and L. Li, Comput. Structures 40, 419 (1991), DOI: 10.1016/0045-7949(91)90367-U. Crossref, Web of Science, Google Scholar
I. Babuška and L. Li, Comput. Methods Appl. Mech. Engrg. 100, 249 (1992), DOI: 10.1016/0045-7825(92)90185-M. Crossref, Web of Science, Google Scholar
C. Bernardi and Y. Maday , Handbook of Numerical Analysis V , eds. P. G. Ciarlet and J. L. Lions ( Elsevier , 1997 ) . Crossref, Google Scholar
C. Chen, Asymptotic convergence rates for the Kirchhoff plate model, Ph.D. Dissertation, The Pennsylvania State University (1995) . Google Scholar
P. G. Ciarlet , Theory of Plates , Mathematical Elasticity, Vol. II ( Elsevier , 1997 ) . Google Scholar
M. Dauge and I. Gruais, Asymptotic Analysis 13, 167 (1996). Crossref, Web of Science, Google Scholar
M. Dauge, I. Gruais and A. Rössle, SIAM J. Math. Anal. 31, 305 (2000), DOI: 10.1137/S0036141098333025. Crossref, Web of Science, Google Scholar
M. Dorr, SIAM J. Numer. Anal. 21, 1180 (1984), DOI: 10.1137/0721073. Crossref, Web of Science, Google Scholar
M. Dorr, SIAM J. Numer. Anal. 23, 58 (1986), DOI: 10.1137/0723005. Crossref, Web of Science, Google Scholar
C. O. Horgan and J. K. Knowles, Adv. Appl. Mech. 23, 179 (1983), DOI: 10.1016/S0065-2156(08)70244-8. Crossref, Web of Science, Google Scholar
R. B. Kellogg, Notes on piecewise smooth elliptic boundary value problems, Institute for Physical Science and Technology, Technical Note BN-1137 (1992) . Google Scholar
K. H. Lo, R. M. Christensen and E. M. Wu, J. Appl. Mech. 46, 663 (1977). Web of Science, Google Scholar
A. L. Madureira, Asymptotics and hierarchical modeling of thin plates, Ph.D. Dissertation, The Pennsylvania State University (1999) . Google Scholar
W. G. Mazja , S. A. Nazarow and B. A. Plamenewski , Asymptotische Theorie Elliptischer Randwertaufgaben in Singulär gestörten Gebieten I ( Akademie Verlag , 1991 ) . Google Scholar
B. Miara, Appl. Anal. 31, 291 (1989), DOI: 10.1080/00036818908839832. Crossref, Google Scholar
D. S. Mitrinović , J. E. Pečarić and A. M. Fink , Inequalities Involving Functions and their Integrals and Derivatives ( Kluwer , 1991 ) . Crossref, Google Scholar
O. Ovaskainen and J. Pitkäranta, SIAM J. Appl. Math. 58, 999 (1998). Web of Science, Google Scholar
C. Schwab, Hierarchic Modelling in Mechanics, eds. M. Ainsworthet al. (Oxford Univ. Press, 1997) pp. 85–160. Google Scholar
M. Vogelius and I. Babuška, Math. Comput. 37, 31 (1981). Web of Science, Google Scholar
M. Vogelius and I. Babuška, Math. Comput. 37, 47 (1981). Web of Science, Google Scholar
M. Vogelius and I. Babuška, Math. Comput. 37, 361 (1981). Crossref, Web of Science, Google Scholar