Hierachical basis preconditioners for second order elliptic problems in three dimensions

Technical Report · Sun Oct 01 04:00:00 EDT 1989

DOI:https://doi.org/10.2172/5005434· OSTI ID:5005434

Go Ong, M. E.

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.

• Research Organization: Washington Univ., Seattle, WA (USA). Dept. of Applied Mathematics
• Sponsoring Organization: DOD; DOE/ER; NSF
• DOE Contract Number: FG06-88ER25061
• OSTI ID: 5005434
• Report Number(s): DOE/ER/25061-5; ON: DE90005980; CNN: AFSOR86-0154; ASC-8519353
• Country of Publication: United States
• Language: English

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