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Title: On the Design and Analysis of Coarse Spaces for Domain Decomposition Methods.


Abstract not provided.

Publication Date:
Research Org.:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
Report Number(s):
DOE Contract Number:
Resource Type:
Resource Relation:
Conference: Proposed for presentation at the Domain Decomposition: Past, Present and Future -- A Workshop in Honor of Olof Widlund's Retirement held February 24-25, 2017 in New York, New York, U.S.A..
Country of Publication:
United States

Citation Formats

Dohrmann, Clark R. On the Design and Analysis of Coarse Spaces for Domain Decomposition Methods.. United States: N. p., 2017. Web.
Dohrmann, Clark R. On the Design and Analysis of Coarse Spaces for Domain Decomposition Methods.. United States.
Dohrmann, Clark R. Wed . "On the Design and Analysis of Coarse Spaces for Domain Decomposition Methods.". United States. doi:.
title = {On the Design and Analysis of Coarse Spaces for Domain Decomposition Methods.},
author = {Dohrmann, Clark R.},
abstractNote = {Abstract not provided.},
doi = {},
journal = {},
number = ,
volume = ,
place = {United States},
year = {Wed Feb 01 00:00:00 EST 2017},
month = {Wed Feb 01 00:00:00 EST 2017}

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  • This paper discusses numerical methods for solving partial differential equations. The topics discussed in this report are: Domain decomposition, Schwarz methods, numerical Schwarz methods, and coarse grid acceleration. 7 refs., 2 tabs. (LSP)
  • Incomplete domain decomposition preconditioning for parallel implementation of the conjugate gradient-like methods is applied to solve the two-group, three-dimensional, coarse mesh finite differenced neutron diffusion equation on the PARAGON XP/S-10 parallel computer. The linear system resulting from implicit time differencing of the time-dependent neutron diffusion equation is solved by the preconditioned biconjugate gradient squared method without employing the fission source iteration. An efficient domain decomposition preconditioning scheme is constructed by taking advantage of strong diagonal dominance of the coarse mesh finite difference formulation. Simplifications are made in the incomplete LU factorization process to construct a preconditioner for a three dimensionalmore » subdomain and the coupling between subdomains is approximated by incorporating only the effect of the non-leakage terms of neighboring subdomains. The method is applied to quarter core and full core fixed source problems which are created from the IAEA three-dimensional benchmark problem. Results show that on a single processor the computation time for the preconditioned biconjugate gradient method is comparable to other conventional iteration methods such as Line-SOR and the cyclic Chebyshev semi-iterative method. The effectiveness of the incomplete domain decomposition preconditioning on a multi-processor is evidenced by the small increase in the number of iterations as the number of subdomains increases. Speedups up to 32.1 are achievable with 64 processing elements for a 34{times}34{times}36 full core three-dimensional problem.« less
  • In recent years, it has turned out that many modern iterative algorithms (multigrid schemes, multilevel preconditioners, domain decomposition methods etc.) for solving problems resulting from the discretization of PDEs can be interpreted as additive (Jacobi-like) or multiplicative (Gauss-Seidel-like) subspace correction methods. The key to their analysis is the study of certain metric properties of the underlying splitting of the discretization space V into a sum of subspaces V{sub j}, j = 1{hor_ellipsis}, J resp. of the variational problem on V into auxiliary problems on these subspaces. Here, the author proposes a modified approach to the abstract convergence theory of thesemore » additive and multiplicative Schwarz iterative methods, that makes the relation to traditional iteration methods more explicit. To this end he introduces the enlarged Hilbert space V = V{sub 0} x {hor_ellipsis} x V{sub j} which is nothing else but the usual construction of the Cartesian product of the Hilbert spaces V{sub j} and use it now in the discretization process. This results in an enlarged, semidefinite linear system to be solved instead of the usual definite system. Then, modern multilevel methods as well as domain decomposition methods simplify to just traditional (block-) iteration methods. Now, the convergence analysis can be carried out directly for these traditional iterations on the enlarged system, making convergence proofs of multilevel and domain decomposition methods more clear, or, at least, more classical. The terms that enter the convergence proofs are exactly the ones of the classical iterative methods. It remains to estimate them properly. The convergence proof itself follow basically line by line the old proofs of the respective traditional iterative methods. Additionally, new multilevel/domain decomposition methods are constructed straightforwardly by now applying just other old and well known traditional iterative methods to the enlarged system.« less