Mathematical Sciences: Domain Decomposition for Time-Dependent Problems

这个项目将研究区域分解 求解含时偏微分的算法 并行处理计算机上的方程。 域 要检查的分解方法是显式/隐式的 Galerkin和混合有限元，有限差分 程序. 在这些方法中，计算域是 划分为不重叠的子域，边界数据在 子域界面的计算显式从解决方案 在前一个时间步。 先验误差估计 这些格式是针对模型问题导出的。 此外，初步的数值结果表明， 方法是可行的，并且加速因子等于 可以实现多个子域名。 主要研究者 我将研究这些算法的扩展， 多重空间中的一般线性和非线性问题 尺寸，以及它们在并行处理上的实现 机械. 特别令人感兴趣的将是应用 这些程序在多孔介质中流动的问题，如 提高石油采收率和地下污染物输送。 在 这些问题，物理和化学过程， 模型通常发生在很长一段时间内，并且准确 模拟需要精细尺度的时间和空间分辨率。 目前的超级计算机在以下方面非常有用： 增加了可获得的分辨率的量。 但据 随着并行计算机的出现，我们可能 能够将分辨率提高至少另一个数量级， 大小为了实现这一目标，需要修改， 现有算法和新算法的发展。 在 方法提出这里，区域分解是用来 有效地将大问题分解为更小的子问题， 可以并行解决。 通过采用这种方法， 精细尺度，多维模拟，目前也 传统机器的存储密集型或耗时的功能可 变得温顺

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.