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

这个项目包括继续研究有限单元法在约束和不适定变分问题上的应用。这包括由研究者和马里兰大学的Douglas Arnold教授开发的一种新的各向异性弹性方程混合有限元方法的研究。重点研究了该方法在纯牵引和混合边界条件下的计算效率。研究者已经为这个难题开发了一些先验误差估计，现在想要在计算机上检查该方法对各种边界条件选择的有效性。最有可能遵循的程序是一个简化的变分方程，它可以简化为一个具有正定矩阵的线性代数方程系统。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问题的有限元方法。所提出的研究包括继续研究使用高于一阶的非一致性有限元（在三角形中点连续的分段多项式），包括推导先验误差估计和计算验证。本研究的思想也将应用于确定抛物型偏微分方程组中未知泛函系数的问题。这一问题的研究与地下油藏的识别有关。这项研究属于数值分析和计算数学的一般领域。其最终目标是发展有效的计算方法，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.