Novel Discretization Schemes for Fully Nonlinear Partial Differential Equations

本项目的目的是开发、分析和实现完全非线性二阶偏微分方程组的有限元方法。这项研究是基于PI的一个最新发现，即拉格朗日有限元方法和间断Galerkin方法可以用来逼近Monge-Ampere方程，这是典型的完全非线性的二阶偏微分方程组。由于这些方法实现简单，因此可以高效、准确地进行高度非线性问题的计算。该项目将在这些结果的基础上进行扩展，以获得一类一般完全非线性方程的简单、高效但准确的数值格式。此外，PI还将开发和分析各种离散化方法，包括混合有限元方法、局部不连续Galerkin方法、杂交Galerkin方法和Petrov Galerkin方法。数学建模在研究和理解自然科学、社会科学和工程中的许多现象方面发挥着关键作用。然而，即使对于简单的问题，封闭形式的解也是不可用的，因此它们的数值近似是唯一可行的选择。随着问题变得越来越复杂，对新的计算方法和创新分析的需求变得迫切，才能让美国在科学和工程方面走在前列。这个项目中研究的这类问题出现在许多数学建模应用中，包括天气现象、确定宇宙的初始形状、最优反射面设计、微分几何、最优传输、数学金融、图像处理和网格生成。尽管它们在物理科学以及纯数学和应用数学中具有重要意义，但这些问题的数值近似仍然是一个相对未被触及的领域。因此，越来越需要为这些类型的方程开发精确的格式。由于解决这些应用问题的进度在很大程度上取决于其控制方程的求解进度，而且这些方程的数值方法仍处于起步阶段，因此在设计、实现和收敛分析方面的任何进展都将对这些应用领域的推进产生直接影响。

The aim of this project is to develop, analyze, and implement the finite element method for fully nonlinear second order partial differential equations (PDEs). The research is based on a recent discovery of the PI that Lagrange finite element methods and discontinuous Galerkin methods can be used to approximate the Monge-Ampere equation, the prototypical fully nonlinear second order PDE. As these methods are simple to implement, the computation of the highly nonlinear problem can be performed efficiently and accurately. The project will expand on these results to obtain simple, efficient, yet accurate numerical schemes for a general class of fully nonlinear equations. In addition, the PI will develop and analyze various discretization methods including mixed finite element methods, local discontinuous Galerkin methods, hybridizable Galerkin methods, and Petrov Galerkin methods.Mathematical modeling plays a key role in the investigation and understanding of many phenomena occurring in the natural sciences, the social sciences and engineering. Yet even for simple problems, closed form solutions are unavailable, and therefore their numerical approximations are the only viable option. As the problems become ever more complex, the need for novel computational methods and innovative analysis becomes imperative to put the United States in the forefront in science and engineering. The class of problems studied in this project arise in numerous mathematical modeling applications including weather phenomena, determining the initial shape of the universe, optimal reflector design, differential geometry, optimal transport, mathematical finance, image processing, and mesh generation. Despite their significance in the physical sciences and pure and applied mathematics, the numerical approximation of these problems remains a relatively untouched area. Therefore, there is a growing need to develop accurate schemes for these types of equations. As progress of solving any of these application problems largely depends on progress of solving their governing equations, and since numerical methods for these equations are still in their infancy, any progress in the design, implementation, and convergence analysis will have an immediate impact in advancing these application areas.