Numerical Methods and Algorithms for Second Order Fully Nonlinear Partial Differential Equations

二阶完全非线性偏微分方程组(PDE)广泛存在于科学和工程领域，如微分几何、最优控制、质量输运、材料科学、气象学、地转流体动力学等。它们构成了一类最难解析和数值逼近的微分方程。在过去的二十年里，基于粘性溶液理论的二阶完全非线性偏微分方程组的理论分析取得了巨大的进展。另一方面，与偏微分方程组分析的成功相比，一般二阶完全非线性偏微分方程组的数值解大多是未被触及的领域，而计算二阶完全非线性偏微分方程组的粘性解是不可行的。在这个研究项目中，PI计划基于新发展的矩解概念和建设性的消失矩方法，对二阶完全非线性偏微分方程组的数值方法和算法进行广泛而系统的研究。该项目的具体任务包括：(I)继续发展Monge-Ampere型偏微分方程组和一般二阶完全非线性椭圆型和抛物型偏微分方程组的矩解理论；(Ii)发展有限元、混合有限元、间断Galerkin和谱Galerkin离散化方法；(Iii)分析所有提出的离散化方法的收敛和收敛速度；(Iv)设计预条件牛顿型非线性求解器；(V)开发基于COMSOL多物理平台的计算机代码，用于在高性能工作站上实现所提出的离散化方法和求解算法。该项目的完成将对二阶完全非线性偏微分方程组的理论研究和数值逼近产生深远的影响。它将为逼近二阶完全非线性偏微分方程组提供第一种实用而成功的方法/途径，这种方法和途径有严格的偏微分方程组和数值理论的支持。作为一种副产品，矩理论将丰富目前对粘性溶液理论的理解，并很有可能为粘性溶液概念提供一个合乎逻辑的、自然的推广。拟议的研究成果将为正确和有效地计算那些来自微分几何、广义相对论、流体力学、材料科学、最优控制、质量运输、气象学、图像处理等领域的完全非线性偏微分方程组提供急需的能力和使能工具，特别是在没有理论的情况下。这个项目的教育部分是让研究生参与并培训他们发展必要的应用和计算数学知识和技能，以便他们能够在未来的科学和工程领域取得成功。

Second order fully nonlinear partial differential equations (PDEs) arise from many areas in science and engineering such as differential geometry, optimal control, mass transportation, materials science, meteorology, geostrophic fluid dynamics. They constitute the most difficult class of differential equations to analyze analytically and to approximate numerically. In the past two decades, enormous advances in the theoretical analysis has been achieved, based on the viscosity solution theory, for second order fully nonlinear PDEs. On the other hand, in contrast to the success of the PDE analysis, numerical solutions for general second order fully nonlinear PDEs is mostly an untouched area, and computing viscosity solutions of second order fully nonlinear PDEs has been impracticable. In this research project, the PI plans to conduct an extensive and systematic study of numerical methods and algorithms for second order fully nonlinear PDEs based on a newly developed moment solution concept and a constructive vanishing moment methodology. The specific tasks of the project include (i) to continue developing the moment solution theory for Monge-Ampere type PDEs and for general second order fully nonlinear elliptic and parabolic PDEs in two and three dimensions; (ii) to develop finite element, mixed finite element, discontinuous Galerkin, and spectral Galerkin discretization methods; (iii) to analyze convergence and rates of convergence for all proposed discretization methods; (iv) to design preconditioned Newton type nonlinear solvers; (v) to develop computer code based on Comsol Multiphysics platform for implementing the proposed discretization methods and solution algorithms on high performance workstations.The completion of the proposed project will have a profound impact on both theoretical study and numerical approximations of second order fully nonlinear PDEs. It will provide the first practical and successful methodology/approach, which is backed by rigorous PDE and numerical theories, for approximating second order fully nonlinear PDEs. As a by-product, the moment solution theory will enrich the current understanding of the viscosity solution theory, and might be very likely to provide a logical and natural generalization/extension for the viscosity solution concept. The findings of the proposed research will provide the much needed capability and enabling tools for computing correctly and efficiently those challenging fully nonlinear PDEs from differential geometry, general relativity, fluid mechanics, materials science, optimal control, mass transportation, meteorology, image processing, especially, in the cases where there are no theories. The educational component of this project is to engage and train graduate students in developing necessary applied and computational mathematics knowledge and skills so that they can pursue a successful career in science and engineering in the future.