Mixed finite elements and smooth approximations for partial differential equations

本项目旨在发展、分析和改进弹性力学方程和Monge-Ampere型方程的数值方法。线弹性力学方程的方法研究将集中在混合有限元方法的改进上。最近在三角形和四面体网格上发展了具有弱强加对称性的非常简单的单元，但这些单元还没有扩展到四边形、二维和三维矩形和六面体网格，这些网格通常是从业者所青睐的。这一建议将使用构造分段多项式精确序列的技术在上述网格上发展稳定的混合有限元。第二个研究领域是Monge-Ampere型方程光滑逼近的构造和分析，以及所发展的方法在解决涉及Monge-Ampere型方程的科学和工程问题中的应用。将遵循几种方法，包括全局优化方法。众所周知，Monge-Ampere型方程和其他完全非线性偏微分方程组一样，不具有一般的光滑解，但一些近似格式，如消失矩方法，需要在光滑函数空间中工作。重点将集中在由研究者和其他人发展的样条元方法的实现和改进上，该方法使用多元样条法来求解高阶偏微分方程组。由于该方法是一致的，所以它具有灵活、稳健、高效和精确的近似，便于实现，对不同元素使用不同次数的多项式的灵活性，以及后验误差估计的简单性。物理现象的数学建模已经成为研究科学和工程中众多问题的标准工具。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，发展数值方法及其分析对这一过程是至关重要的。这个项目涉及在基本问题中出现的两种类型的方程，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。Monge-Ampere型方程出现在各种几何和变分问题中，例如Monge-Kantorovich问题。它们还出现在气象学、流体力学、非线性弹性、材料科学和数学金融等应用领域。来自该项目的新方法的开发有可能将更具竞争力的工具交到国家科学家和工程师手中。该项目的教育部分是，它将向新一代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，有助于维护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.