Mathematical Sciences: Domain Decomposition for Time-Dependent Problems

本项目将研究在并行处理计算机上求解含时偏微分方程组的区域分解算法。要研究的区域分解方法有显式/隐式Galerkin和混合有限元，以及有限差分程序。在这些方法中，计算域被划分为不重叠的子域，子域交界处的边界数据由前一时间步长的解显式计算。对于模型问题，我们得到了这些格式的先验误差估计。此外，初步的数值结果表明，该方法是可行的，并且可以获得与子域数目相等的加速因子。主要研究人员将研究将这些算法扩展到更一般的多维空间中的线性和非线性问题，以及它们在并行处理机上的实现。特别令人感兴趣的是将这些程序应用于在多孔介质中流动的问题，例如提高石油采收率和地下污染物传输。在这些问题中，要模拟的物理和化学过程通常发生在很长的时间段内，而精确的模拟需要精细的时间和空间分辨率。目前的超级计算机在提高可获得的分辨率方面发挥了极其重要的作用。然而，随着新兴的并行计算机的出现，我们可能至少能够将分辨率提高另一个数量级。要实现这一目标，需要对现有算法进行修改，并开发新的算法。在本文提出的方法中，区域分解被用来有效地将大问题分解成可以并行求解的较小的子问题。通过采用这种类型的方法，当前对于传统机器来说过于耗费内存或时间的精细、多维模拟可能变得容易处理。

This project will investigate domain decomposition algorithms for solving time-dependent partial differential equations on parallel processing computers. The domain decomposition methods to be examined are explicit/implicit Galerkin and mixed finite element, and finite difference procedures. In these approaches, the computational domain is divided into nonoverlapping subdomains, and boundary data at subdomain interfaces are calculated explicitly from the solution at the previous time step. A priori error estimates for these schemes have been derived for model problems. Furthermore, preliminary numerical results indicate that the approaches are viable, and that a speed-up factor equal to the number of subdomains can be achieved. The principal investigator will investigate the extension of these algorithms to more general linear and nonlinear problems in multiple space dimensions, and their implementation on parallel processing machines. Of particular interest will be the application of these procedures to flow in porous media problems, such as enhanced oil recovery and subsurface contaminant transport. In these problems, the physical and chemical processes to be modeled generally occur over long time periods, and accurate simulation requires fine-scale temporal and spatial resolution. Current supercomputers have been extremely useful in increasing the amount of resolution obtainable. However, it is possible that with the emerging parallel computers, we may be able to increase resolution by at least another order of magnitude. To achieve this goal will require modifications to current algorithms and the development of new algorithms. In the methods proposed here, domain decomposition is used to effectively divide large problems into smaller subproblems that can be solved in parallel. By employing this type of approach, fine-scale, multidimensional simulations that are currently too memory -intensive or time-consuming for conventional machines may become tractable.