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Conjugate gradient method for dual-dual mixed formulations


Authors: Gabriel N. Gatica and Norbert Heuer
Journal: Math. Comp. 71 (2002), 1455-1472
MSC (2000): Primary 65N30, 65N22, 65F10
DOI: https://doi.org/10.1090/S0025-5718-01-01394-1
Published electronically: December 5, 2001
MathSciNet review: 1933040
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Abstract: We deal with the iterative solution of linear systems arising from so-called dual-dual mixed finite element formulations. The linear systems are of a two-fold saddle point structure; they are indefinite and ill-conditioned. We define a special inner product that makes matrices of the two-fold saddle point structure, after a specific transformation, symmetric and positive definite. Therefore, the conjugate gradient method with this special inner product can be used as iterative solver. For a model problem, we propose a preconditioner which leads to a bounded number of CG-iterations. Numerical experiments for our model problem confirming the theoretical results are also reported.


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Additional Information

Gabriel N. Gatica
Affiliation: GI$^{2}$MA, Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile.
Email: ggatica@ing-mat.udec.cl

Norbert Heuer
Affiliation: GI$^{2}$MA, Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile.
Email: norbert@ing-mat.udec.cl

DOI: https://doi.org/10.1090/S0025-5718-01-01394-1
Keywords: Mixed finite elements, dual-dual variational formulation, conjugate gradient method
Received by editor(s): September 22, 1999
Received by editor(s) in revised form: October 3, 2000
Published electronically: December 5, 2001
Additional Notes: This research was partially supported by CONICYT-Chile through Program A on Numerical Analysis of the FONDAP in Applied Mathematics and Fondecyt project No. 1980122, and by the Dirección de Investigación of the Universidad de Concepción through the Advanced Research Groups Program
Article copyright: © Copyright 2001 American Mathematical Society

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