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Title: Hierachical basis preconditioners for second order elliptic problems in three dimensions

Abstract

The discretization uses nodal basis functions and the preconditioner arises. We present preconditioners for a symmetric, positive definite linear system arising from the finite element discretization of a second order elliptic problem in three dimensions. The discretization uses nodal basis functions and the preconditioner arises from the use of hierarchical basis functions. We show that the condition number of the linear hierarchical basis coefficient matrix {cflx A} scaled by a coarse grid operator is O(N{sup 1/3} log N{sup 1/3}) when uniform tetrahedral refinement is used, where N is the number of unknowns. If additional diagonal scaling by levels is applied in the fine grid, a condition number of O(N{sup 1/3}) is obtained. The same result is obtained if {cflx A} is scaled by its block diagonal. Moreover, we show that any other block diagonal scaling of {cflx A} will yield a condition number that grows at least as O(N{sup 1/3}). These results compare favorably with the condition number of O(N{sup 2/3}) of the nodal coefficient matrix. We provide numerical results that confirm this theory. These results are extensions of those obtained by Yserentant for two dimensional problems. We also extend the analysis of the linear preconditioner to the case ofmore » non-uniform refinement. 74 refs., 78 figs., 14 tabs.« less

Authors:
Publication Date:
Research Org.:
Washington Univ., Seattle, WA (USA). Dept. of Applied Mathematics
Sponsoring Org.:
USDOD; DOE/ER; National Science Foundation (NSF)
OSTI Identifier:
5005434
Report Number(s):
DOE/ER/25061-5
ON: DE90005980; CNN: AFSOR86-0154; ASC-8519353
DOE Contract Number:  
FG06-88ER25061
Resource Type:
Technical Report
Country of Publication:
United States
Language:
English
Subject:
99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; ELLIPTICAL CONFIGURATION; BOUNDARY-VALUE PROBLEMS; ALGORITHMS; CONVERGENCE; FINITE ELEMENT METHOD; HILBERT SPACE; ITERATIVE METHODS; PARALLEL PROCESSING; THREE-DIMENSIONAL CALCULATIONS; BANACH SPACE; CONFIGURATION; MATHEMATICAL LOGIC; MATHEMATICAL SPACE; NUMERICAL SOLUTION; PROGRAMMING; SPACE; 990200* - Mathematics & Computers

Citation Formats

Go Ong, M. E. Hierachical basis preconditioners for second order elliptic problems in three dimensions. United States: N. p., 1989. Web. doi:10.2172/5005434.
Go Ong, M. E. Hierachical basis preconditioners for second order elliptic problems in three dimensions. United States. https://doi.org/10.2172/5005434
Go Ong, M. E. 1989. "Hierachical basis preconditioners for second order elliptic problems in three dimensions". United States. https://doi.org/10.2172/5005434. https://www.osti.gov/servlets/purl/5005434.
@article{osti_5005434,
title = {Hierachical basis preconditioners for second order elliptic problems in three dimensions},
author = {Go Ong, M. E.},
abstractNote = {The discretization uses nodal basis functions and the preconditioner arises. We present preconditioners for a symmetric, positive definite linear system arising from the finite element discretization of a second order elliptic problem in three dimensions. The discretization uses nodal basis functions and the preconditioner arises from the use of hierarchical basis functions. We show that the condition number of the linear hierarchical basis coefficient matrix {cflx A} scaled by a coarse grid operator is O(N{sup 1/3} log N{sup 1/3}) when uniform tetrahedral refinement is used, where N is the number of unknowns. If additional diagonal scaling by levels is applied in the fine grid, a condition number of O(N{sup 1/3}) is obtained. The same result is obtained if {cflx A} is scaled by its block diagonal. Moreover, we show that any other block diagonal scaling of {cflx A} will yield a condition number that grows at least as O(N{sup 1/3}). These results compare favorably with the condition number of O(N{sup 2/3}) of the nodal coefficient matrix. We provide numerical results that confirm this theory. These results are extensions of those obtained by Yserentant for two dimensional problems. We also extend the analysis of the linear preconditioner to the case of non-uniform refinement. 74 refs., 78 figs., 14 tabs.},
doi = {10.2172/5005434},
url = {https://www.osti.gov/biblio/5005434}, journal = {},
number = ,
volume = ,
place = {United States},
year = {Sun Oct 01 00:00:00 EDT 1989},
month = {Sun Oct 01 00:00:00 EDT 1989}
}