Adaptive FEM for controlling pointwise errors and level sets

椭圆型偏微分方程在科学和工程应用中普遍存在，其快速精确的数值求解一直是一个重要的研究领域。 自适应有限元算法，自动构建高效的计算网格是非常流行的工具来解决这样的方程。 自适应有限元法（AFEM）是一种迭代反馈过程，其中计算初始近似解，然后使用来自初始近似的信息来构造更好的近似。 这里所谓的“更好的近似”取决于计算的期望输出。 大多数自适应码被设计为控制误差的能量范数（一阶导数的均方根或平均值），因为能量范数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式有关，并且通常不能保证能量误差的控制将导致误差的其他测量的计算有效控制。 因此，最近的许多研究都集中在使用自适应代码来计算“感兴趣的量”无关的能量norm.This项目涉及的自适应算法的建设和分析的计算目标量是相关的逐点信息的错误，或者是在整个计算域的位置。 需要这种信息的情况包括在热平衡时定位物体中的最高温度，以及确定弹性体中的应力超过给定阈值的位置。 水平集方法，其中不断发展的接口表示为水平集的解决方案偏微分方程也得到普及，在最近几年。 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几个方面的一个基本理论的后验估计的逐点误差在简单的模型问题。 这一理论将得到丰富和推广。 其次，我们将研究这一基本理论在应用中的重要系统，特别是来自流体动力学和线性弹性方程的定常Stokes系统中的应用。 最后，我们将开发一个自适应算法，严格控制水平集的解决方案，椭圆问题。 拟议的研究还提供了一个研究生在数值分析和科学计算的培训。

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.