ALGORITHMS: Scalable Solvers for Nonlinear Partial Differential Equations

这项为期三年的研究工作的重点是设计、分析和软件实现一类并行非线性迭代方法，用于数值求解一些在计算流体力学和计算生物学中具有重要应用的高度非线性偏微分方程组。工程中要考虑的非线性偏微分方程组通常不是强椭圆的，它们往往含有非椭圆分量，从而导致解的非光滑和局部奇性，如边界层或尖锐的锋面。对于光滑的非线性问题，传统的非线性方法，如牛顿方法，能够以接近二次收敛的速度减少全局非线性，但一旦局部奇异性出现在计算域的某个地方，即使这种情况只发生在大的线性系统的几个分量上，也会变得非常慢。该算法基于Cai和Kyes在2001年提出的一类求解具有不平衡非线性的代数非线性方程的非线性预条件算法，在非线性预条件中，将全局问题分解为子问题，并对子问题进行子空间非线性消除。然后通过Schwarz型区域分解方法将子问题粘合在一起。由于子空间非线性消除，消除了局部奇异性，从而使全局系统具有更均匀的非线性。将使用多层/多网格、区域分解、非线性预处理和非线性消除方法的组合来研究一系列这样的算法。粗略地说，在这些算法中，区域分解提供了并行性，多层提供了关于并行计算机上的问题大小和处理器数量的可扩展性，非线性消除消除了对局部奇异性的敏感性。为了研究算法在高性能计算机上的并行性能，如工作站和超级计算机的集群，将开发一个与Argonne国家实验室的PETSC完全互操作的插件库。所提出的算法和软件开发将对应用领域产生重大影响，也将对计算科学中其他需要求解大型非线性方程的领域产生实质性影响。

The focus of this three-year research effort is the design, analysis and software implementation of a class of parallel nonlinear iterative methods for the numerical solution of some highly nonlinear partial differential equations (PDEs) arising from important applications in computational fluid dynamics and computational biology. The nonlinear PDEs to be considered in the project are usually not strongly elliptic, and they often contain non-elliptic components causing the solution to be nonsmooth and have local singularities, such as boundary layers or sharp fronts. For smooth nonlinear problems, traditional nonlinear methods, such as Newton's methods, are capable of reducing the global nonlinearities at a nearly quadratic convergence rate but become very slow once the local singularities appear somewhere in the computational domain, even if this happens to only a few components of a largenonlinear system. The proposed algorithm is motivated by the class of nonlinear preconditioning algorithms introduced by Cai and Keyes in 2001 for solving algebraic nonlinear equations that have unbalanced nonlinearities.In nonlinear preconditioning, the global problem is partitioned into subproblems, and subspace nonlinear eliminations are performed on all subproblems. The subproblem are then 'glued' together by a Schwarz type domain decomposition method. Due to the subspace nonlinear elimination, the local singularities are removed, and the global system therefore has more uniform nonlinearity. A family of such algorithms will be studied using a combination of multilevel/multigrid, domain decomposition, nonlinear preconditioning, and nonlinear elimination methods. Roughly speaking, in these algorithms, domain decomposition provides the parallelism, multilevel provides the scalability with respect to the problem size and to the number of processors on parallel computers, and nonlinear elimination removes the sensitivity to the local singularities.Several important application problems will be considered, including the steady state incompressible Navier-Stokes equations with high Reynolds number and the optimization of a steady state biofluid problem. To study the parallel performance of the algorithms on high performance computers, such as a cluster of workstations and supercomputers, a library will be developed as a plug-in package that is fully interoperable with PETSc of Argonne National Laboratory. The proposed algorithm and software development will have a great impact on the application areas, and will also have substantial influence on other areas of computational sciences where large nonlinear equations need to be solved.