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

这个项目涉及有限元法的继续研究 约束和不适定变分问题。这包括 方程的一种新的混合有限元方法的研究 各向异性弹性的研究者和教授 马里兰州大学的道格拉斯阿诺德。重点主要放在 纯牵引和混合牵引方法的计算有效性 边界条件调查者产生了一些先验错误 估计这个难题，现在想检查 该方法在计算机上的各种选择的有效性， 边界条件最有可能遵循的程序是 一个简化的变分方程，它可化为一个线性方程组， 具有正定矩阵的代数方程。为了解决这个系统 将有效地使用预处理共轭梯度法。 另一个项目涉及有限元方法， 一阶标量双曲问题拟议的研究包括 继续调查使用的三维有限元， 阶数高于一阶（分段多项式，在 三角形的中点），包括先验误差的推导 估计和计算验证。本研究的思路是 也将适用于识别未知功能的问题， 抛物型偏微分方程组中的系数。 这一问题的研究与地下油藏有关 识别. 本研究福尔斯属于数值分析的一般范畴， 计算数学其最终目标是发展高效 计算方法用于科学问题， 空间分布的质量、能量、应力和流。

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.