Nonoverl Apping Domain Decomposition Finite Difference Methods For Parabolic Problems

Mathematical physics and engineering problems can be turned into the problems of solving large scale partial differential equations, such as reservoir simulation, the design of large scale of spacecraft structure engineering, aerodynamics, reactor, etc. There exists the same difficulty in effectively solving the huge-scale computation with little error. As is well known, domain decomposition is one powerful way to solve this problem in parallel, which is potentially more efficient in both time and storage. The primary technique consists of solving subproblems on various subdomains, while enforcing suitable continuity requirements between adjacent subproblems, till the local solutions converge (within a specified accuracy) to the true solution. Domain decomposition methods are of most interest when applied to discretization of the differential equaitons (either by finite difference, finite element, spectral or spectral element methods), mostly classified as either an overlapping or a nonoverlapping subdomain approach. The earliest known iterative domain decomposition technique is proposed in the pioneering work of H. A. Schwarz in 1870 to prove the existence of harmonic functions on irregular regions which are the union of overlapping subregions. Variants of Schwarz's method are later studied in the 1950s by Sobolev. Babuska. Courant and Hilbert et al., and the recent interest in domain decomposition is initiated in studies by Dryja, Bramble, Lions, Meurant, Chan and the others to develop the inherent parallelism in the 1980s. Domain decomposition has been widely used in computational mathematics and engineering science and other applied fields [32, 54, 55, 57, 58. 59, 60. 61].Nonoverlapping domain decomposition algorithms are important due to high efficiency, adaptability for models and flexibility for domain discretization, based on a decomposition of the whole domain into various nonoverlapping subdomains. Preconditioners at interface boundary must be discussed for these methods, usually considered in the following three continuous forms: the Dirichlet-Dirichlet (D-D), the Neumann-Neumann (N-N). and the Dirichlet-Neumann (D-N). Much work has been studied by George, Liu, Dryja. Smith and...