ALGORITHMS: Scalable Solvers for Nonlinear Partial Differential Equations

这个为期三年的研究工作的重点是一类并行非线性迭代方法的设计，分析和软件实现的数值解的一些高度非线性偏微分方程（PDE）所产生的重要应用在计算流体力学和计算生物学。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，导致解是非光滑的，并具有局部奇异性，如边界层或尖锐的前沿。对于光滑的非线性问题，传统的非线性方法，如牛顿法，能够以接近二次的收敛速度减少全局非线性，但一旦局部奇异性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组件上，也会变得非常缓慢。该算法是在Cai和Keyes于2001年提出的求解非平衡非线性代数方程组的非线性预处理算法的基础上发展起来的.子问题，然后“粘”在一起的施瓦茨型区域分解方法。由于子空间非线性消除，局部奇异性被去除，因此全局系统具有更一致的非线性。一个家庭这样的算法将使用多级/多重网格，区域分解，非线性预处理，非线性消除方法的组合进行研究。粗略地说，在这些算法中，区域分解提供了并行性，多级提供了关于问题大小和并行计算机上的处理器数目的可伸缩性，而非线性消除消除了对局部奇异性的敏感性。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上的并行性能，如工作站和超级计算机集群，将开发一个库作为一个插件包，与阿贡国家实验室的PETSc完全互操作。本文提出的算法和软件开发将对应用领域产生巨大的影响，也将对需要求解大型非线性方程的计算科学的其他领域产生实质性的影响。

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.