hp-adaptive FOSLS Methods for Nonlinear PDE Problems with Singularities

本提案旨在开发一种有效的hp自适应一阶系统最小二乘（FOSLS）方法，用于可能表现出奇异性和其他非光滑性的非线性偏微分方程（PDEs）。通用的FOSLS方法已经成为解决科学和工程中基于pde的问题的有力工具。它的一个主要优点是，它可以用来将一组给定的方程转换成一个松散耦合的标量方程组，这种方程组可以很容易地被多层有限单元处理。重新表述是通过将方程重铸为最小二乘原理与精心导出的一阶系统相关联来完成的。FOSLS的另一个优点是，相关的函数本身提供了一个自然的尖锐的局部误差估计器，可以用于有效的自适应细化。在之前的工作中，FOSLS方法被开发和分析用于线性弹性、线性输运和不可压缩流体动力学等问题。在原PDE的标准h2型平滑假设下，从理论上和数值上证明了FOSLS具有最优的有限元和多网格收敛特性。然而，目前有两个主要的困难阻碍了FOSLS有效地应用于更复杂的问题：非光滑解和非线性。该项目的目标是开发一个自适应的加权函数和自适应嵌套迭代方案来解决这些困难。目标是获得一种方案，该方案可以求解复杂的非线性PDE系统，其总成本相当于最细网格上的少量松弛步骤。这个项目的成功完成将增强我们在高端计算平台上模拟复杂物理过程的能力。特别是，该项目解决耦合流体/结构相互作用的计算模型，例如，在生物力学系统中，如顺应血管中的血流。其他可能从这个项目中受益的领域包括空气动力学、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工作来解决一个单一的复杂问题，开发最佳的数值算法变得更加重要。这个项目的成功完成将扩大存在最优算法的问题类别。

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.