Mathematical Sciences: Finite Element Methods for Partial Differential Equations

本项目继续平板模型的数值和分析研究，并研究近似线性双曲方程和对流扩散方程的有限元方法。第一个研究领域是Reissner-Mindlin板模型，该模型确定了板中部的横向位移和垂直于板中部的纤维的旋转，作为具有适当边界条件的耦合偏微分方程组的解。与双调和板模型不同，该模型有一个边界层，通过发展和严格证明解在板厚的幂次方面的渐近展开来研究。研究者希望展示边界层的强度如何取决于模型中使用的特定物理边界条件，以及在某些边界条件下，在边界的平坦部分附近不存在边界层。利用这些结果，他将开发有限元近似方案，在区域内部提供更高的收敛速度，并且可以通过针对边界层确切性质的网格细化技术提高总体收敛速度。他还打算研究由纯位移或混合变分原理导出的高阶模型。他将通过渐近展开研究解析性质（如边界层效应），并开发不受“锁定”影响的近似方案，“锁定”是Reissner-Mindlin模型的一个常见问题，会导致薄板精度下降。由于所有这些模型都是全三维模型的近似值，因此本研究的一个重要方面是比较各种模型对三维解的性质做出的预测。最后，该项目涉及的推导局部误差估计的有限元方法近似线性双曲和对流扩散方程。这样做的目的是为了证明近似格式的相关域性质与精确解的相关域性质非常相似。板模型是一种二维数学模型，它大大简化了对薄的三维弹性体的研究。当施加不同的力时，它们通常被工程师用来预测物体的位移和应力。该项目旨在使用严格的数学分析来更好地理解模型的预测，开发改进的计算算法，用于在数值上近似组成模型的偏微分方程，并比较板块模型对三维物体特性的预测。特别地，对每个模型所预测的边界层现象进行了严格的研究。（边界层是靠近物体边界的区域，在该区域内各种物理量发生快速变化。）最后，推导了在各种应用中用作数学模型的线性双曲方程和对流扩散方程的有限元法的“局部”误差估计，将严格地表明精确解的某些重要定性性质与有限元法计算的近似解非常接近。这使该方法的使用者相信，数值结果可靠地预测了所模拟的物理量的行为。

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.