Mathematical Sciences: Finite Element Methods for Partial Differential Equations

该项目涉及发展和 偏微分有限元分析 由力学产生的方程。 研究课题 包括以下步骤： (1)边界层的研究 Reissner-Mindlin板模型，包括发展和 严格的理由的渐近展开 板厚适当幂的解 对于边界条件建模的“软”简支 免费盘子 (2)高阶一致精度有限差分格式的发展 Reissner-Mindlin模型的单元方法（推广了 一阶方法以前开发的）。 此外该 本文将用（1）中发展的边界层理论来分析 方法的内部行为，并开发网格细化 提高准确性的策略。 (3)有限元法应用的探讨 弹性力学中的边值问题。 特别 注意Korn第二不等式的有效性 对于具有牵引边界条件的可伸缩单元 并确定有限元方法之间的关系 基于线性方程组的等效公式， 弹性 (4)三维有限元法的应用研究 一阶线性标量的逼近方法 双曲方程 一个应用是解决 识别椭圆偏微分方程中的变系数 微分方程

This project is concerned with the development and analysis of finite element methods for partial differential equations arising from mechanics. The topics of research include: (1) Investigation of the boundary layer of the Reissner-Mindlin plate model, including the development and rigorous justification of an asymptotic expansion of the solution in terms of appropriate powers of the plate thickness for boundary conditions modelling the "soft" simply supported and free plate. (2) The development of higher order uniformly accurate finite element methods for the Reissner-Mindlin model (generalizing a first order method previously developed). In addition, the boundary layer theory developed in (1) will be used to analyze the interior behavior of methods and to develop mesh refinement strategies for improved accuracy. (3) Investigation of the use of nonconforming finite elements for boundary value problems in elasticity. Particular attention is given to the validity of Korn's second inequality for nonconforming elements with traction boundary conditions and to determining the relationship of finite element methods based on equivalent formulations of the equations of linear elasticity. (4) Investigation of the use of nonconforming finite element methods for the approximation of first order linear scalar hyperbolic equations. One application is to the problem of identifying a variable coefficient in an elliptic partial differential equation.