Mathematical Sciences: Finite Element Methods for Partial Differential Equations

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This project continues the numerical and analytical investigation of plate models and also studies finite element methods to approximate linear hyperbolic and convection-diffusion equations. The first area of study deals with the Reissner-Mindlin plate model, which determines the transverse displacement of the midplane and the rotation of the fibers normal to the midplane of a plate as the solution of a coupled system of partial differential equations with appropriate boundary conditions. Unlike the biharmonic plate model, this model has a boundary layer, which is studied by developing and rigorously justifying an asymptotic expansion of the solution in terms of powers of the plate thickness. The investigator expects to show how the strength of the boundary layer depends on particular physical boundary conditions used in the model and that for certain boundary conditions no boundary layer exists near a flat portion of the boundary. Using these results, he will develop finite element approximation schemes that give higher rates of convergence in the interior of the domain and for which the overall convergence rate may be enhanced by mesh refinement techniques geared to the exact nature of the boundary layer. He also intends to study higher order models, derived from pure displacement or mixed variational principles. He will investigate analytical properties (such as boundary layer effects) by means of asymptotic expansions and develop approximation schemes that do not suffer from "locking," a common problem for the Reissner-Mindlin model that causes deterioration in accuracy for thin plates. Because all these models are approximations to the full three-dimensional model, an important aspect of this research is to compare the predictions the various models make about properties of the three-dimensional solution. Finally, the project involves the derivation of local error estimates for a finite element method for the approximation of linear hyperbolic and convection-diffusion equations. The intent is to show that the domain of dependence properties of the approximation scheme closely imitate those of the exact solution. %%% Plate models are two-dimensional mathematical models that greatly simplify the study of thin three-dimensional elastic bodies. They are commonly used by engineers to predict displacements and stresses of objects when various forces are applied. The project aims to use a rigorous mathematical analysis to better understand the predictions of the models, to develop improved computational algorithms for numerically approximating the partial differential equations that comprise the model, and to compare the predictions the plate models make about properties of the three-dimensional object. In particular, a rigorous study is made of the boundary layer phenomena predicted by each model. (The boundary layer is a region near the boundary of the object in which various physical quantities undergo rapid changes.) Finally, the derivation of "local" error estimates for a finite element method to approximate linear hyperbolic and convection-diffusion equations, which are used as mathematical models in a variety of applications, will rigorously show that certain important qualitative properties of the exact solution are closely imitated by the approximate solution computed by the finite element method. This gives confidence to the users of the method that the numerical results reliably predict the behavior of the physical quantities being modeled.