A Time-Dependent Dirichlet-Neumann Method for the Heat Equation

We present a waveform relaxation version of the Dirichlet-Neumann method for parabolic problem. Like the Dirichlet-Neumann method for steady problems, the method is based on a non-overlapping spatial domain decomposition, and the iteration involves subdomain solves with Dirichlet boundary conditions followed by subdomain solves with Neumann boundary conditions. However, each subdomain problem is now in space and time, and the interface conditions are also time-dependent. Using a Laplace transform argument, we show for the heat equation that when we consider finite time intervals, the Dirichlet-Neumann method converges, similar to the case of Schwarz waveform relaxation algorithms. The convergence rate depends on the length of the subdomains as well as the size of the time window. In this discussion, we only stick to the linear bound. We illustrate our results with numerical experiments.Comment: 9 pages, 5 figures, Lecture Notes in Computational Science and Engineering, Vol. 98, Springer-Verlag 201

Domain Decomposition Algorithms Of Schwarz Type, Designed For Massively Parallel Computers

We discuss implementation of additive Schwarz type algorithms on SIMD computers. A recursive, additive algorithm is compared with a two-level scheme. These methods are based on a subdivision of the domain into thousands of micro-patches that can reflect local properties, coupled with a coarser, global discretization where the `macro' behavior is reflected. The two-level method shows very promising flexibility, convergence and performance properties when implemented on a massively parallel SIMD computer