Mathematical Sciences: Finite Element Methods for Partial Differential Equations

本项目涉及力学偏微分方程的有限元方法的发展和分析。研究课题包括：(1) Reissner-Mindlin板模型边界层研究；包括在模拟“软”简支和自由板的边界条件下，根据板厚的适当幂的解的渐近展开的发展和严格的证明。(2)开发了Reissner-Mindlin模型的高阶均匀精度有限元方法（推广了先前开发的一阶方法）。此外，(1)中发展的边界层理论将用于分析方法的内部行为，并制定网格细化策略以提高精度。(3)研究非协调有限元在弹性边值问题中的应用。特别注意到对于具有牵引边界条件的非协调单元的Korn第二不等式的有效性，并根据线性弹性方程的等效公式确定有限元方法之间的关系。(4)非协调有限元法在一阶线性标量双曲方程逼近中的应用研究。一个应用是识别椭圆型偏微分方程中的可变系数的问题。

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.