Mathematical Sciences: Finite Element Methods for Constrained and Ill-Posed Variational Problems

这个项目包括对约束和不适定变分问题的有限元方法的持续研究。这包括研究人员和马里兰大学的道格拉斯·阿诺德教授开发的一种新的各向异性弹性方程的混合有限元方法。重点讨论了该方法在纯牵引力和混合边界条件下的计算有效性。研究人员已经为这个困难的问题开发了一些先验误差估计，现在想在计算机上检查该方法在各种边界条件选择下的有效性。最有可能遵循的过程是简化的变分方程，它归结为具有正定矩阵的线性代数方程组。为了有效地求解该系统，将使用预条件共轭梯度法。另一个要进行的项目涉及一阶标量双曲型问题的有限元方法。这项研究包括继续研究高阶非协调有限元(在三角形中点连续的分段多项式)的使用，包括先验误差估计的推导和计算验证。本文的研究思路也将应用于抛物型偏微分方程组中未知泛函系数的辨识问题。这一问题的研究对于地下油藏的识别具有重要意义。这项研究属于数值分析和计算数学的一般领域。它的最终目标是开发有效的计算方法，用于涉及空间分布的质量、能量、应力和流动的科学问题。

This project involves continued research on finite element methods for constrained and ill-posed variational problems. This includes research on a new mixed finite element method for the equations of anisotropic elasticity developed by the investigator and Professor Douglas Arnold of the University of Maryland. The focus is mainly on the computational effectivness of the method for pure traction and mixed boundary conditions. The investigator has developed some a priori error estimates for this difficult problem and now wants to check the effectiveness of the method on a computer for various choices of boundary conditions. The most likely procedure that will be followed is a simplified variational equation which reduces to a system of linear algebraic equations with a positive definite matrix. To solve the system efficiently, a preconditioned conjugate gradient method will be used. Another project to be pursued involves finite element methods for the first order scalar hyperbolic problem. The proposed research consists of continued investigations on the use of nonconforming finite elements of order higher than one (piecewise polynomials which are continuous at the midpoints of triangles) including the derivation of a priori error estimates and computational verification. The ideas of this research will also be applied to the problem of identifying unknown functional coefficients in a parabolic system of partial differential equations. The work on this problem is relevant to underground oil reservoir identification. This research falls into the general area of numerical analysis and computational mathematics. Its ultimate goal is to develop efficient computational methods to be used in scientific problems involving spatially distributed masses, energies, stresses and flows.