Meshfree Finite Difference Methods for Nonlinear Elliptic Equations

非线性椭圆方程描述了各种各样的问题，如透镜和反射器的设计、地球地下的测绘、医学图像的解释和复杂天气现象的建模。然而，现有的求解这些方程的工具仅在非常简单的情况下是实用的，并且在面对现实数据时可能会失败。本课题将介绍一类求解非结构化非光滑非线性椭圆方程的新方法。在这个项目中开发的新的数学和计算技术将导致快速、可靠的方法来解决当前应用中进一步发展所需的现实环境中的方程。本课题将介绍一类新的求解二维和三维非线性退化椭圆方程的无网格有限差分方法。鉴于现有的完全非线性方程的收敛方法通常需要在矩形域的均匀网格上进行计算，该框架将允许在非结构化点云上提出方程。方法将依赖于异常大的搜索邻域，以构建与底层PDE操作符结构一致的近似值。所得到的方案将正确地近似弱（粘度）解，同时允许自适应和复杂的几何形状。该项目还将引入几种非经典边界条件的新公式，这些公式将用于产生无网格实现。所得到的代数系统的快速求解技术将得到发展。

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.