Mathematical Sciences: Finite Element Methods for Partial Differential Equations

本项目致力于力学偏微分方程组有限元方法的发展和分析。研究内容包括：(1)Reissner-Mindlin平板模型边界层的研究，包括推导和严格证明了在模拟“软”简支和自由板的边界条件下，关于板厚度的适当幂的解的渐近展开式。(2)发展了Reissner-Mindlin模型的高阶一致精度有限元方法(推广了以前发展的一阶方法)。此外，(1)中发展的边界层理论将被用来分析方法的内部行为，并开发网格细化策略以提高精度。(3)弹性力学边值问题的非协调有限元应用研究。重点讨论了Korn第二个不等式对于具有牵引力边界条件的非协调单元的有效性，以及基于线弹性力学方程的等价形式确定有限元方法之间的关系。(4)研究了用非协调有限元方法逼近一阶线性标量双曲型方程。其中一个应用是识别椭圆型偏微分方程中的变量系数的问题。

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.