Mathematical Sciences: Finite Element Methods for Partial Differential Equations

该项目继续数值和分析 板模型的研究，并研究有限元 线性双曲型和对流扩散型近似方法 方程 第一个研究领域涉及 Reissner-Mindlin板模型，它确定了横向 中平面的位移和纤维的旋转 垂直于板的中平面作为耦合的解 适当的偏微分方程组 边界条件 与双谐波板模型不同，这个 模型有一个边界层，这是通过发展和 严格证明解的渐近展开式， 板厚的幂项。 调查人员预计， 来说明边界层的强度是如何取决于 模型中使用的特定物理边界条件， 对于某些边界条件，不存在边界层 靠近边界的平坦部分。 利用这些结果，他 将开发有限元逼近方案，给出 更高的收敛速度在内部的域和为 通过网格可以提高整体收敛速度 适合于边界的精确性质的精化技术 层.他还打算研究高阶模型，来自 纯位移或混合变分原理。 他将 研究分析性质（如边界层 效果）的渐近展开和发展 近似方案不遭受“锁定”，一个共同的 Reissner-Mindlin模型的问题，导致恶化 在薄板的精确度上。 因为所有这些模型 一个重要的三维模型， 这项研究的一个方面是比较各种预测 模型对三维解的性质的影响。 最后，本项目涉及到局部误差的推导 的有限元方法的估计 线性双曲型和对流扩散方程。 意图 是要表明，域的依赖性质的 近似方案密切模仿的精确解。 %%% 板块模型是二维数学模型 这大大简化了薄的三维弹性的研究， 尸体 它们通常被工程师用来预测 物体的位移和应力 应用。 该项目旨在使用严格的数学 分析，以更好地理解模型的预测， 开发改进的计算算法， 近似偏微分方程，包括 模型，并比较板块模型的预测 三维物体的属性。 特别是， 对边界层现象进行了严格的研究 每个模型的预测。 (The边界层是靠近 物体的边界，其中各种物理量 迅速变化）。 最后，推导出“局部”误差 有限元方法逼近线性 双曲和对流扩散方程，它们被用作 数学模型在各种应用中，将严格 表明，某些重要的定性性质的确切 近似解与解的相似性 用有限元法计算。 这给了信心， 用户认为该方法的数值结果可靠 预测被建模的物理量的行为。

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.