Novel Discretization Schemes for Fully Nonlinear Partial Differential Equations

本项目的目的是开发、分析和实现全非线性二阶偏微分方程的有限元方法。该研究是基于最近π的一个发现，即拉格朗日有限元方法和不连续伽辽金方法可以用来近似Monge-Ampere方程，这是典型的完全非线性二阶偏微分方程。由于这些方法实现简单，可以高效、准确地计算高度非线性问题。该项目将在这些结果的基础上进行扩展，以获得一般类型的全非线性方程的简单、有效而准确的数值格式。此外，PI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。数学建模在调查和理解自然科学、社会科学和工程中发生的许多现象方面起着关键作用。然而，即使是简单的问题，封闭形式的解决方案是不可用的，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，为了使美国在科学和工程领域走在前列，对新颖计算方法和创新分析的需求变得势在必行。在这个项目中研究的一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、最佳反射器设计、微分几何、最佳运输、数学金融、图像处理和网格生成。尽管它们在物理科学、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触及的领域。因此，越来越需要为这些类型的方程制定精确的格式。由于解决任何这些应用问题的进展在很大程度上取决于解决其控制方程的进展，并且由于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推进这些应用领域产生直接影响。

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.