HIERARCHICAL MODELING OF THE HEAT EQUATION IN A HETEROGENEOUS PLATE ANA CAROLINA CARIUS ALEXANDRE L. MADUREIRA Abstract. In this work we investigate the modeling of heterogeneous plates, where the length scale of the heterogeneity can be much smaller than the area of the plate’s middle surface. We derive a two-dimensional model for the original problem, and the resulting PDEs not only have rough coefficients but also depend on the thickness, resulting in a singularly perturbed problem. We employ asymptotic techniques to show that, as the plate thickness tends to zero, our model converges to the exact solution. To tame the numerical troubles of the resulting model we use finite elements methods of multiscale type. 1. Introduction The challenge of solving PDEs in beams, plates and shells has historically attracted re- searchers from different fields, not only because of the importance of the physical problems demanding such task, but also because of the beautiful problems arising from the endeavor. Focusing on plates, the first necessary step is to perform some sort of dimension reduction, i.e., model a three-dimensional problem with a two-dimensional model. Hopefully, the re- sulting equations are easier to solve, and the final solution approximates in some sense the exact solution of the original problem. There are basically three known ways, not always exclusive, to obtain plate models. Proba- bly the most common arguments are based on physical properties of the underlying problem, often combined with some mathematical reasoning. It is also possible to derive the models using asymptotic techniques, usually with a sound mathematical basis, and the results easier Date : December 10, 2007. Key words and phrases. Homogenization, Dimension reduction, Residual Free Bubbles, Multiscale Finite Element Method. The authors would like to thank Gabriel Barrenechea and Fr´ ed´ eric Valentin for the kind invitation to participate in the mini-symposium Stabilized and Multiscale Finite Element Methods, at the Second Chilean Workshop on Numerical Analysis of Partial Differential Equations - WONAPDE 2007. Gabriel and Fred were important influences in the making of this present work. The first author had support from CAPES, and the second author was partially supported by the CNPq/Brazil Projects 306104/2004-0 and 486026/2006-0. 1