hp-adaptive FOSLS Methods for Nonlinear PDE Problems with Singularities

本文提出了一种有效的hp自适应一阶系统最小二乘（FOSLS）方法，用于求解可能具有奇异性和其他非光滑性质的非线性偏微分方程（PDE）。一般FOSLS方法已经成为解决科学和工程中基于偏微分方程问题的有力工具。它的一个主要优点是可以将一组给定的方程转化为一个松散耦合的标量方程组，从而可以很容易地用多层有限元来处理。重新制定是通过重铸的方程作为一个最小二乘原理与仔细推导的一阶系统。FOSLS的另一个优点是，相关的功能本身提供了一个自然的尖锐的局部误差估计器，它可以用于有效的自适应细化。在以前的工作中，FOSLS方法被开发和分析的问题，包括线性弹性，线性传输，和不可压缩流体动力学。在原偏微分方程的H2型光滑性假设下，理论和数值证明了FOSLS具有最优有限元和多重网格收敛特性。然而，两个主要的困难目前阻止FOSLS被有效地应用于更复杂的问题：非光滑的解决方案和非线性。该项目的目的是开发一个自适应加权功能加上自适应嵌套迭代计划来处理这些困难。我们的目标是获得一个方案，可以解决复杂的非线性偏微分方程系统的总成本相当于少量的松弛步骤上的最精细的网格。这个项目的成功完成将提高我们的能力，在高端计算平台上模拟复杂的物理过程。特别是，该项目解决了耦合流体/结构相互作用的计算模型，例如，在生物力学系统中，如顺应性血管中的血流。其他可能从这个项目中受益的领域是空气动力学、天体物理学、地球物理学和气象学。随着现代计算机体系结构变得越来越复杂，利用成千上万的处理器一起处理一个复杂的问题，开发最优数值算法变得更加重要。这个项目的成功完成将拓宽最优算法存在的问题的类别。

This proposal is for development of an effective hp-adaptive First-OrderSystem Least-Squares (FOSLS) method for nonlinear partial differentialequations (PDEs) that may exhibit singularities and other non-smoothness properties. The general FOSLS methodology is already a powerful tool forthe solution of PDE-based problems in science and engineering. One of itsmain advantages is that it can be used to transform a given set ofequations into a loosely coupled system of scalar equations that can betreated easily by multilevel finite elements. The reformulation is doneby recasting the equations as a least-squares principle associated with a carefully derived first-order system. Another advantage of FOSLS isthat the associated functional itself provides a natural sharp local errorestimator, which can be employed for effective adaptive refinement. Inprevious work, the FOSLS methodology was developed and analyzed forproblems that include linear elasticity, linear transport, andincompressible fluid dynamics. Under standard H2-type smoothnessassumptions on the original PDE, FOSLS has been proved theoretically and numerically to exhibit optimal finite element and multigridconvergence properties. However, two principal difficulties currentlyprevent FOSLS from being applied effectively to more complicated problems:nonsmooth solutions and nonlinearity. The project aim is to develop anadaptive weighted-functional coupled with adaptive nested iterationschemes to treat these difficulties. The goal is to obtain a scheme thatcan solve complex nonlinear PDE systems with a total cost equivalent to a small number of relaxation steps on the finest grid.Successful completion of this project will enhance our ability to modelcomplex physical processes on high-end computing platforms. In particular,this project addresses computational models of coupled fluid/structureinteractions that occur, for example, in biomechanical systems such asblood flow in compliant vessels. Other areas that could potentiallybenefit from this project are aerodynamics, astrophysics, geophysics andmeteorology. As modern computer architectures become more complex,harnessing 10s of thousands of processors working together on a single,complex problem, it becomes even more important to develop optimalnumerical algorithms. Successful completion of this project will widen theclass of problems for which optimal algorithms exist.