Adaptive FEM for controlling pointwise errors and level sets

用于控制点误差和水平集的自适应有限元

• 批准号: 0713770
• 负责人: Alan Demlow
• 金额: $ 11.28万
• 依托单位: University of Kentucky Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0713770&HistoricalAwards=false
• 关键词: Adaptive; FEM; controlling; pointwise; errors

Elliptic partial differential equations are ubiquitous in science and engineering applications, and their fast and accurate numerical solution remains an important area of research. Adaptive finite element algorithms for automatically constructing efficient computational grids are very popular tools for solving such equations. An adaptive finite element method (AFEM) is an iterative feedback procedure in which an initial approximate solution is computed, and information from the initial approximation is then used to construct a better approximation. What is meant here by "better approximation" depends upon the desired output from the computation. Most adaptive codes are designed to control the energy norm (root-mean-square, or average, of the first derivatives) of the error because the energy norm is closely associated with the finite element algorithm. The goal output from many computations, on the other hand, is related to other ways of measuring the error, and there is generally no guarantee that control of the energy error will lead to computationally efficient control of other measures of the error. Much recent research has thus focused on using adaptive codes to compute "quantities of interest" not related to the energy norm.This project involves the construction and analysis of adaptive algorithms for computations where the goal quantity is either related to pointwise information about the error, or is a location within the overall computational domain. Situations where such information is desirable include locating the maximum temperature in a body at thermal equilibrium and determining where the stresses in an elastic body exceed a given threshold. Level set methods in which evolving interfaces are represented as level sets of solutions to partial differential equations have also gained popularity in recent years. This project contains three main goals. First, the PI and others have previously developed several aspects of a basic theory for a posteriori estimation of pointwise errors in simple model problems. This theory will be enriched and extended. Secondly, we will investigate application of this basic theory to systems important in applications, in particular the stationary Stokes system from fluid dynamics and equations of linear elasticity. Finally, we will develop an adaptive algorithm that rigorously controls level sets of solutions to elliptic problems. The proposed research also provides for the training of a graduate student in numerical analysis and scientific computing.

椭圆形的部分微分方程在科学和工程应用中无处不在，它们的快速，准确的数值解决方案仍然是研究的重要领域。 自动构建有效计算网格的自适应有限元算法是求解此类方程的非常流行的工具。 自适应有限元方法（AFEM）是一个迭代反馈过程，其中计算初始近似解决方案，然后使用初始近似中的信息来构造更好的近似值。 “更好的近似”在这里的含义取决于计算中所需的输出。 大多数自适应代码旨在控制误差的能量标准（第一位衍生物的根平方或平均值），因为能量标准与有限元算法密切相关。 另一方面，来自许多计算的目标输出与测量误差的其他方法有关，通常不能保证控制能量误差将导致对误差的其他度量的计算有效控制。 因此，许多最新的研究集中在使用自适应代码来计算与能量规范无关的“关注量”。该项目涉及针对计算的自适应算法的构建和分析，在该计算中，目标数量与有关误差的点信息有关，或者是整体计算域内的位置。 所需信息的情况包括在热平衡处定位体内的最高温度，并确定弹性体中的应力超过给定阈值。 近年来，将不断发展的界面表示为部分微分方程的解决方案集合的水平集方法。 该项目包含三个主要目标。 首先，PI和其他人以前已经开发了基本理论的几个方面，用于在简单模型问题中对重点错误的后验估计。 该理论将被丰富和扩展。 其次，我们将研究该基本理论对应用程序中重要的系统的应用，尤其是从线性弹性的流体动力学和方程式的固定stokes系统。 最后，我们将开发一种自适应算法，该算法严格控制椭圆问题的解决方案集合集。 拟议的研究还规定了在数值分析和科学计算中培训研究生的培训。