Preconditioning and a posteriori error estimates using h- and p-hierarchical finite elements with rectangular supports

We show some of the properties of the algebraic multilevel iterative methods when the hierarchical bases of finite elements (FEs) with rectangular supports are used for solving the elliptic boundary value problems. In particular, we study two types of hierarchies; the so‐called h‐ and p‐hierarchical...

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Bibliographic Details
Published in:Numerical linear algebra with applications 2009-05, Vol.16 (5), p.415-430
Main Author: Pultarova, I
Format: Article
Language:English
Subjects:
a posteriori error estimate
algebraic multilevel method
CBS constant
hierarchical finite element basis
hierarchical preconditioning
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Summary:We show some of the properties of the algebraic multilevel iterative methods when the hierarchical bases of finite elements (FEs) with rectangular supports are used for solving the elliptic boundary value problems. In particular, we study two types of hierarchies; the so‐called h‐ and p‐hierarchical FE spaces on a two‐dimensional domain. We compute uniform estimates of the strengthened Cauchy–Bunyakowski–Schwarz inequality constants for these spaces. Moreover, for diagonal blocks of the stiffness matrices corresponding to the fine spaces, the optimal preconditioning matrices can be found, which have tri‐ or five‐diagonal forms for h‐ or p‐refinements, respectively, after a certain reordering of the elements. As another use of the hierarchical bases, the a posteriori error estimates can be computed. We compare them in test examples for h‐ and p‐hierarchical FEs with rectangular supports. Copyright © 2008 John Wiley & Sons, Ltd.
ISSN:1070-5325
1099-1506
DOI:10.1002/nla.624