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，用于在计算流体动力学和计算生物学中重要应用的一些高度非线性偏微分方程（PDEs）的数值解。项目中要考虑的非线性偏微分方程通常不是强椭圆的，它们通常包含非椭圆分量，导致解不光滑并具有局部奇异性，如边界层或锐锋。对于光滑非线性问题，传统的非线性方法，如牛顿方法，能够以接近二次的收敛速度减少全局非线性，但一旦局部奇异点出现在计算域的某个地方，即使这只发生在一个大型非线性系统的几个分量上，也会变得非常缓慢。所提出的算法的动机是由Cai和Keyes在2001年引入的一类非线性预处理算法，用于求解具有不平衡非线性的代数非线性方程。在非线性预处理中，将全局问题划分为子问题，并对所有子问题进行子空间非线性消去。然后通过Schwarz型域分解方法将子问题“粘合”在一起。由于子空间非线性消去，消除了局部奇异点，使全局系统具有更均匀的非线性。一系列这样的算法将使用多层/多网格、域分解、非线性预处理和非线性消除方法的组合来研究。粗略地说，在这些算法中，领域分解提供了并行性，多层提供了与问题大小和并行计算机上的处理器数量有关的可伸缩性，非线性消除消除了对局部奇异性的敏感性。将考虑几个重要的应用问题，包括高雷诺数的稳态不可压缩Navier-Stokes方程和稳态生物流体问题的优化。为了研究算法在高性能计算机（如工作站集群和超级计算机）上的并行性能，将开发一个库作为插件包，与Argonne国家实验室的PETSc完全互操作。所提出的算法和软件开发将对应用领域产生重大影响，也将对需要求解大型非线性方程的其他计算科学领域产生重大影响。

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.