Meshfree Finite Difference Methods for Nonlinear Elliptic Equations

非线性椭圆型方程描述了各种各样的问题，如透镜和反射器的设计，绘制地球地下地图，解释医学图像，以及模拟复杂的天气现象。然而，现有的求解这些方程的工具只在非常简单的情况下才实用，当面对真实的数据时可能会失败。这个项目将介绍一类新的方法，当数据是非结构化和非光滑的时候，求解非线性椭圆型方程。本项目开发的新的数学和计算技术将导致在现实环境中快速、可靠地求解方程，以满足当前应用的进一步发展。本项目将引入一类新的无网格有限差分方法来求解二维和三维非线性退化椭圆型方程。虽然现有的完全非线性方程的收敛方法通常需要在矩形区域中的均匀网格上进行计算，但该框架将允许在非结构化的点云上提出方程。方法将依赖于异常大的搜索邻域，以便构建与底层PDE运算符的结构一致的近似。所得到的格式将正确地逼近弱(粘性)解，同时允许适应性和复杂的几何形状。该项目还将引入几个非经典边界条件的新公式，这些条件将用于产生无网格实施。由此产生的代数系统的快速求解技术将被开发出来。

Nonlinear elliptic equations describe problems as varied as the design of lenses and reflectors, mapping the subsurface of the earth, interpretation of medical images, and modelling complex weather phenomena. However, existing tools for solving these equations are practical only in very simple settings, and can fail when faced with realistic data. This project will introduce a new class of methods for solving nonlinear elliptic equations when the data is unstructured and non-smooth. The new mathematical and computational techniques developed in this project will lead to fast, reliable methods for solving equations in the realistic settings required for further progress in current applications.This project will introduce a new class of meshfree finite difference methods for solving nonlinear degenerate elliptic equations in two- and three-dimensions. Whereas existing convergent methods for fully nonlinear equations often require computations to be performed on a uniform grid in a rectangular domain, this framework will allow equations to be posed on unstructured point clouds. Methods will rely on unusually large search neighborhoods in order to construct approximations that align with the structure of the underlying PDE operator. The resulting schemes will correctly approximate weak (viscosity) solutions, while allowing for adaptivity and complicated geometries. This project will also introduce new formulations of several non-classical boundary conditions, which will be used to produce meshfree implementations. Fast solution techniques for the resulting algebraic systems will be developed.