Mixed finite elements and smooth approximations for partial differential equations

本项目的目标是发展、分析和改进弹性力学方程和Monge-Ampere型方程的数值计算方法，线弹性力学方程的数值计算方法的研究将集中于改进混合有限元方法。最近在三角形和四面体网格上开发了具有弱对称性的非常简单的单元，但这些单元尚未扩展到四边形，2D和3D矩形和六面体网格，这些网格通常受到从业者的青睐。本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。研究的第二个领域是建设和分析的顺利逼近蒙赫-安培型方程和应用的方法开发的解决问题的科学和工程涉及蒙赫-安培型方程。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一样，一般不具有光滑解，但一些近似方案，例如消失矩方法，需要在光滑函数空间中工作。重点将放在样条元素法的实施和改进上，该方法由研究者和其他人开发，它使用多元样条来求解高阶偏微分方程。它导致灵活的，强大的，高效的和准确的近似允许易于实施，使用多项式的不同程度上的不同元素和简单的后验误差估计的灵活性，因为该方法是conformation.Mathematical建模的物理现象已经成为标准的工具，在科学和工程中的许多问题的调查。但通常所得到的方程没有可以用简单数学公式表示的解。因此，发展的数值方法和分析是必不可少的这一进程。该项目解决两种类型的方程出现在基本问题，但这里开发的方法的影响远远超出了正在考虑的特定应用。 弹性力学方程在许多工业、生物和工程应用中出现。Monge-Ampere型方程出现在各种几何和变分问题中，例如Monge-Kantorovich问题。它们也出现在应用领域，如气象学，流体力学，非线性弹性，材料科学和数学金融。该项目开发的新方法有可能为国家科学家和工程师提供更具竞争力的工具。该项目的教育部分是，它将介绍新一代的学生计算数学涉及实际问题。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.