Mixed finite elements and smooth approximations for partial differential equations

该项目旨在发展、分析和改进弹性方程和蒙日-安培型方程的数值方法。线性弹性方程方法的研究将集中在对混合有限元方法的改进上。最近在三角形和四面体网格上开发了非常简单的弱对称元素，但这些元素尚未扩展到从业者经常青睐的四边形，二维和三维矩形和六面体网格上。本提案将使用构造分段多项式精确序列的技术来开发上述网格上的稳定混合有限元。第二个研究领域是蒙日-安培型方程的光滑近似的构造和分析，以及将开发的方法应用于解决涉及蒙日-安培型方程的科学和工程问题。将遵循几种方法，包括全局优化方法。众所周知，Monge-Ampere型方程，像其他全非线性偏微分方程一样，不具有一般的光滑解，但一些近似方案，如消失矩方法，需要在光滑函数空间中工作。重点将放在样条元方法的实施和改进上，该方法是由研究者和其他人开发的，它使用多元样条来解高阶偏微分方程。它导致灵活，稳健，高效和准确的近似，允许易于实现，在不同元素上使用不同程度多项式的灵活性和后验误差估计的简单性，因为该方法是一致的。物理现象的数学建模已经成为研究科学和工程中许多问题的标准工具。但通常得到的方程没有可以用简单的数学公式表示的解。因此，数值方法及其分析的发展对这一过程至关重要。本项目涉及出现在基本问题中的两类方程，但这里开发的方法的影响远远超出了所考虑的特定应用。弹性方程出现在许多工业、生物和工程应用中。蒙日-安培型方程出现在各种几何和变分问题中，如蒙日-坎托洛维奇问题。它们也出现在气象学、流体力学、非线性弹性、材料科学和数学金融等应用领域。该项目开发的新方法有可能为美国的科学家和工程师提供更具竞争力的工具。该项目的教育意义在于，它将向新一代的学生介绍涉及实际问题的计算数学。因此，这也有助于国家安全，并有助于保持国家的全球科学领导地位。

The project is directed towards the development, analysis and improvement of numerical methods for the elasticity equations and Monge-Ampere type equations.The research in methods for the linear elasticity equations will focus on improvement of mixed finite element methods. Very simple elements with weakly imposed symmetry have been recently developed on triangular and tetrahedral meshes but these elements have yet to be extended to quadrilateral, 2D and 3D rectangular and hexahedral meshes which are often favored by practitioners. This proposal will use the technique of constructing piecewise polynomial exact sequences for the development of stable mixed finite elements on the above mentioned meshes. The second area of study is the construction and analysis of smooth approximations to Monge-Ampere type equations and the application of the methods developed to the solution of problems from science and engineering involving Monge-Ampere type equations. Several approaches will be followed, including global optimization ones. As it is well known the Monge-Ampere type equations, like other fully nonlinear partial differential equations do not possess in general smooth solutions but several of the approximation schemes, e.g. the vanishing moment methodology, require to work in spaces of smooth functions. The focus will be on the implementation and improvement of the spline element method, developed by the investigator and others, which uses multivariate splines for the solution of higher order partial differential equations. It leads to flexible, robust, efficient and accurate approximations allowing easy implementation, the flexibility of using polynomials of different degrees on different elements and the simplicity of a posteriori error estimates since the method is conforming.Mathematical modeling of physical phenomena have become the standard tool for the investigation of numerous problems in science and engineering. But often the resulting equations do not have solutions that can be represented by simple mathematical formulas. Hence the development of numerical methods and their analysis is essential to this process. This project adresses two types of equations which appear in fundamental problems but the impact of the methods developed here goes well beyond the particular applications being considered. The elasticity equations appear in many industrial, biological and engineering applications. The Monge-Ampere type equations appear in various geometric and variational problems, e.g.the Monge-Kantorovich problem. They also appear in applied fields such as meteorology, fluid mechanics, nonlinear elasticity, material sciences and mathematical finance. The development of the new methods from this project have the potential to put more competitive tools in the hands of the nation's scientists and engineers. The educational component of the project is that it will introduce a new generation of students to computational mathematics involving practical problems. Therefore this also contributes to national security and helps maintain the global scientific leadership position of the nation.