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• W2022397154 abstract "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 $hat{A}$ preconditioned by a coarse grid operator is $O(h^{-1} log h^{-1})$, where h is the mesh size. If additional diagonal scaling by levels is applied in the fine grid, a condition number of $O(h^{-1})$ is obtained. The same result is obtained if $hat{A}$ is preconditioned by its block diagonal. Moreover, any other block diagonal preconditioning of $hat{A}$ will yield a condition number that grows at least as $O(h^{-1})$. These results compare favorably with the condition number of $O(h^{-2})$ 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." @default.
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• W2022397154 date "1997-03-01" @default.
• W2022397154 modified "2023-09-24" @default.
• W2022397154 title "Hierarchical Basis Preconditioners in Three Dimensions" @default.
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• W2022397154 doi "https://doi.org/10.1137/s1064827594276539" @default.
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