Nonlinear Domain Decomposition Methods and Software for Multicomponent Problems

研究的重点是一类并行非线性区域分解方法的设计和软件实现的数值解的高度非线性多分量偏微分方程所产生的重要应用在计算流体动力学和计算生物学。算法和软件开发将对几个重要的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计算科学和工程的其他领域产生重大影响。为了扩大研究的影响，该软件将与广泛使用的PETSc软件包完全兼容。PETSc的当前用户，以及具有PETSc接口的其他软件包的用户，将能够访问算法和软件。这些多分量问题通常不是椭圆的，并且这些非椭圆分量经常导致解具有局部奇异性，例如边界层或尖锐前沿。这些新算法是由求解非平衡非线性代数方程组的非线性消元算法和非线性预处理算法发展而来的。在新算法中，首先识别出尺度不一致的非线性分量，然后利用微积分的隐函数定理将这些分量替换为其余尺度更一致的分量的函数。因此，原始系统被简化为只包含光滑分量。被认为是几个重要类的应用程序，包括在传统的CFD和计算生物学的新兴领域的问题。为了研究并行性能的算法在高性能计算机上的库将开发作为一个插件程序包，是完全可互操作的PETSc的阿贡国家实验室。

The focus of the research is the design and software implementation of a class of parallel nonlinear domain decomposition methods for the numerical solution of highly nonlinear multicomponent partial differential equations arising from important applications in computational fluid dynamics and computational biology. The algorithms and software development will have a great impact on several important application areas, and will also have substantial influence on other areas of computational sciences and engineering where large nonlinear equations need to be solved. To broaden the impact of the research, the software will be made fully compatible with the widely used PETSc package. Current users of PETSc, as wellas users of other packages with interfaces to PETSc, will be able to access to the algorithms and software.These multicomponent problems are usually not elliptic, and these non-elliptic components often cause the solution to have local singularities, such as boundary layers or sharp fronts. The new algorithms are motivated by the class of nonlinear elimination algorithms and nonlinear preconditioning algorithms for solvingalgebraic nonlinear equations with unbalanced nonlinearities. In the new algorithms the misscaled nonlinear components are first identified, then using the implicit function theorem of calculus these components are replaced by a function of the remaining more uniformly scaled components. The original system is hence reducedimplicitly to contain only the smooth components. Several important classes of applications are considered including problems in traditional CFD and problems in the emerging field of computational biology. To study the parallel performance of the algorithms on high performance computers a library will be developed as a plug-inpackage that is fully interoperable with PETSc of Argonne National Laboratory.