Mathematical Sciences: Finite Element Methods For Incompressible, Viscous Flows

研究者研究高雷诺数下粘性不可压缩流的求解算法。这个问题包括：边界层和内层、主导非线性和敏感非线性、高度非对称和可能无限的线性系统和不可压缩约束的相互关联的困难。从一个方面到另一个方面研究了解决这些困难的算法。（例如，以增加求解非线性的难度为代价来简化线性系统是不令人满意的。）一般的方法是(1)高阶方法(2)点自适应和非结构化网格，(3)稳定的有限元离散化，(4)非线性的稳定，多层次，多步牛顿方法，(5)线性系统的鲁棒（=均匀），并行，迭代求解器，以及(6)对所有算法发展的充分数学支持。高雷诺数的流体流动问题出现在许多技术和科学应用中，例如材料凝固过程中熔化区域的对流、空气和地下水中污染物的输送和扩散以及气候变化的模拟。由于这些方程的精确解是不可能的，因此基于计算机的流体流动问题模拟对于准确预测和最终控制感兴趣的量是必不可少的。高雷诺数流动问题的准确、可靠模拟是本课题研究的一个极具挑战性的科学问题。在大规模应用中，这涉及到对高度并行的算法的研究。

Layton The investigator studies solution algorithms for viscous, incompressible flows at high Reynolds number. This problem includes the interrelated difficulties of: boundary and interior layers, dominating and sensitive nonlinearities, highly nonsymmetric and possibly indefinite linear systems and the incompressibility constraint. Solution algorithms are studied for each of these difficulties from one aspect to another. (For example, it would not be satisfactory to simplify the linear system at the expense of increasing the difficult of resolving the nonlinearity.) The general approach is (1) higher order methods (2) point adaptivity and unstructured meshes, (3) stabilized finite element discretizations, (4) stabilized, multi-level, multi-step Newton methods for the nonlinearity, (5) robust (= uniform in Re), parallel, iterative solvers for the linear systems, and (6) full mathematical support for all algorithmic developments. Fluid flow problems at high Reynolds number arise in many technological and scientific applications, such as convection in the melted region in the solidification of materials, transport and dispersion of pollutants in air and groundwater and simulations of climatic changes. Since exact solution of these equations is impossible, computer based simulation of fluid flow problems is essential in accurately predicting, and ultimately controlling the quantities which are of interest. Accurate and reliable simulation of high Reynolds number flow problems is a very challenging scientific problem which is studied in this research. In large scale applications this involves the study of algorithms which are highly parallel.