Mathematical Sciences: Domain Decomposition for Time-Dependent Problems

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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.