Hierarchical modeling of the heat equation in a heterogeneous plate

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.