Topics in Mathematical Theory of Adaptive Finite Element Methods

自适应有限元方法数学理论专题

基本信息

批准号：
1720369
负责人：
Alan Demlow
金额：
$ 18.02万
依托单位：
Texas A&M University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-09-01 至 2021-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1720369&HistoricalAwards=false
关键词：
Topics Mathematical Theory Adaptive Finite

项目摘要

Finite element methods (FEM) are widely used to approximately solve partial differential equations in simulations of physical phenomena arising in engineering and the physical sciences. Such simulations are an indispensable tool in the development and testing of new technologies. Adaptive variants of finite element methods are designed to increase the efficiency and accuracy with which simulations can be carried out by making better use of computational resources and to increase confidence in the accuracy of simulations by providing researchers with a computable measure of the errors that arise in approximation techniques. This research project aims to develop new variants of adaptive finite element methods and increase mathematical understanding of their underpinnings. The project has two main foci. The first is adaptive FEM for partial differential equations defined on surfaces, which arise for example in describing fluid flows with multiple components (such as oil and water). The second is development and analysis of adaptive FEM for controlling various measures of the error, especially maximum errors.In the first project the investigator will construct and analyze adaptive variants of surface finite element methods with two main goals in mind. First, while surface FEM are an established finite element methodology with many useful variants defined, adaptive versions of some important variants are missing. This project aims to fill that gap. Secondly, the project will explore the interaction between adaptive surface FEM, the way a given surface is represented in a finite element code, and the smoothness or regularity of the surface. The result will be more robust adaptive surface codes that give users greater flexibility in representing surfaces while also making the best possible use of available information about the surface. The second main project will lead to proof of convergence of adaptive algorithms for controlling maximum errors, and will also provide new adaptive algorithms for controlling maximum errors in a class of singularly perturbed elliptic problems.

有限元方法（FEM）被广泛用于求解工程和物理科学中物理现象的模拟中的部分微分方程。这些模拟是开发和测试新技术的必不可少的工具。有限元方法的自适应变体旨在通过更好地利用计算资源来提高模拟的效率和准确性，并通过为研究人员提供对近似技术中出现的误差的可计算量度，从而提高对模拟准确性的信心。该研究项目旨在开发自适应有限元方法的新变体，并增加对其基础的数学理解。该项目有两个主要焦点。第一个是对表面上定义的部分微分方程的自适应FEM，例如在描述具有多个组分（例如油和水）的流体流中。第二个是自适应FEM的开发和分析，用于控制误差的各种措施，尤其是最大错误。在第一个项目中，研究者将构建和分析具有两个主要目标的表面有限元方法的自适应变体。首先，尽管表面FEM是已建立的有限元方法，并且定义了许多有用的变体，但缺少某些重要变体的自适应版本。该项目旨在填补这一空白。其次，该项目将探索自适应表面FEM之间的相互作用，在有限元代码中表示给定表面的方式以及表面的平滑度或规律性。结果将是更强大的自适应表面代码，可为用户提供更大的灵活性来表示表面，同时还可以最好地使用有关表面的可用信息。第二个主要项目将导致自适应算法收敛以控制最大错误，还将提供新的自适应算法，以控制一类奇异扰动的椭圆问题中的最大错误。