Hierarchical basis preconditioners in three dimensions

Journal Article · Sat Mar 01 04:00:00 EST 1997 · SIAM Journal on Scientific Computing

DOI:https://doi.org/10.1137/S1064827594276539· OSTI ID:471155

Ong, M E.G. ^[1]

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1. Univ. of California, San Diego, CA (United States). Dept. of Mathematics

This paper presents hierarchical basis preconditioners for symmetric positive definite linear systems arising from the finite element discretization of second-order elliptic problems in three dimensions. The problem is discretized using nodal basis functions, and the preconditioner arises from a transformation to hierarchical basis functions. For the case where tetrahedral elements and quasi-uniform refinement are used, the condition number of the linear hierarchical basis coefficient matrix {cflx A} preconditioned by a coarse grid operator is O(h{sup {minus}1} log h{sup {minus}1}), where h is the mesh size. If additional diagonal scaling by levels is applied in the fine grid, a condition number of O(h{sup {minus}1}) is obtained. The same result is obtained if {cflx A} is preconditioned by its block diagonal. Moreover, any other block diagonal preconditioning of {cflx A} will yield a condition number that grows at least as O(h{sup {minus}1}). These results compare favorably with the condition number of O(h{sup {minus}2}) of the nodal coefficient matrix. Numerical results that illustrate the theory are provided. The sequential implementation of this preconditioner in three dimensions using tetrahedral elements takes only 4n operations per iteration, where n is the number of unknowns.

• Sponsoring Organization: National Science Foundation, Washington, DC (United States); USDOE, Washington, DC (United States)
• DOE Contract Number: FG03-87ER25037; W-7405-ENG-48
• OSTI ID: 471155
• Journal Information: SIAM Journal on Scientific Computing, Journal Name: SIAM Journal on Scientific Computing Journal Issue: 2 Vol. 18; ISSN 1064-8275; ISSN SJOCE3
• Country of Publication: United States
• Language: English

Related Subjects

99 GENERAL AND MISCELLANEOUS
FINITE ELEMENT METHOD
MATRICES
MESH GENERATION
PARTIAL DIFFERENTIAL EQUATIONS
TRANSFORMATIONS