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Meshfree Finite Difference Methods for Nonlinear Elliptic Equations

非线性椭圆方程的无网格有限差分法

基本信息

批准号：
1619807
负责人：
Brittany Froese Hamfeldt
金额：
$ 15万
依托单位：
New Jersey Institute of Technology
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2016
资助国家：
美国
起止时间：
2016-09-01 至 2021-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1619807&HistoricalAwards=false
关键词：
Meshfree Finite Difference Methods Nonlinear

项目摘要

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.

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，绘制地球的地下地图，医学图像的解释，以及模拟复杂的天气现象。然而，用于求解这些方程的现有工具仅在非常简单的设置中是实用的，并且在面对现实数据时可能会失败。本项目将介绍一类新的方法，用于解决非结构化和非光滑数据时的非线性椭圆方程。本项目开发的新的数学和计算技术将导致在当前应用进一步发展所需的现实环境中求解方程的快速、可靠的方法。本项目将介绍一类新的无网格有限差分方法，用于求解二维和三维的非线性退化椭圆方程。而现有的完全非线性方程的收敛方法通常需要在矩形域中的均匀网格上进行计算，该框架将允许方程在非结构化点云上进行。方法将依赖于异常大的搜索邻域，以构建与底层PDE算子的结构一致的近似。由此产生的计划将正确地近似弱（粘性）的解决方案，同时允许自适应和复杂的几何形状。本项目还将介绍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。快速解决所产生的代数系统的技术将被开发。