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Multigrid and multilevel methods for nonconforming $Q_1$ elements

Authors: Zhangxin Chen and Peter Oswald
Journal: Math. Comp. 67 (1998), 667-693
MSC (1991): Primary 65N30, 65N22, 65F10
MathSciNet review: 1451319
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Abstract: In this paper we study theoretical properties of multigrid algorithms and multilevel preconditioners for discretizations of second-order elliptic problems using nonconforming rotated $Q_1$ finite elements in two space dimensions. In particular, for the case of square partitions and the Laplacian we derive properties of the associated intergrid transfer operators which allow us to prove convergence of the $\mathcal {W}$-cycle with any number of smoothing steps and close-to-optimal condition number estimates for $\mathcal {V}$-cycle preconditioners. This is in contrast to most of the other nonconforming finite element discretizations where only results for $\mathcal {W}$-cycles with a sufficiently large number of smoothing steps and variable $\mathcal {V}$-cycle multigrid preconditioners are available. Some numerical tests, including also a comparison with a preconditioner obtained by switching from the nonconforming rotated $Q_1$ discretization to a discretization by conforming bilinear elements on the same partition, illustrate the theory.

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Additional Information

Zhangxin Chen
Affiliation: Department of Mathematics, Box 156, Southern Methodist University, Dallas, Texas 75275–0156
MR Author ID: 246747

Peter Oswald
Affiliation: Institute of Algorithms and Scientific Computing, GMD - German National Research Center for Information Technology, Schloß Birlinghoven, D-53754 Sankt Augustin, Germany

Keywords: Finite elements, mixed methods, nonconforming methods, multigrid methods, multilevel preconditioners, elliptic problems
Received by editor(s): December 21, 1995
Received by editor(s) in revised form: November 11, 1996
Additional Notes: The first author is partly supported by National Science Foundation grant DMS-9626179.
Article copyright: © Copyright 1998 American Mathematical Society