Bramble-Pasciak-type conjugate gradient method for Darcýs equations

Faisal Fairag; Mohammed Alshahrani; Hattan Tawfiq

doi:10.1137/151003143

Bramble-Pasciak-type conjugate gradient method for Darcýs equations

Faisal Fairag, Mohammed Alshahrani, Hattan Tawfiq

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

We consider the solution of a system of linear algebraic equations which is obtained from a Raviart-Thomas mixed finite element formulation of Darcýs equations. In [C. E. Powell and D. Silvester, SIAM J. Matrix Anal. Appl., 25 (2003), pp. 718-738], Powell and Silvester developed a block-diagonal preconditioner for this system. In this research work, we extend their results by constructing a block-triangular preconditioner and establish an eigenvalue bound for the preconditioned matrix. The preconditioned matrix is nonsymmetric but it is self-adjoint in a nonstandard inner product. The Bramble-Pasciak-type conjugate gradient method is used to solve the linear system which ensures that the norm of the error is minimized. Numerical experiments confirm the theoretical results and illustrate good convergence properties.

Original language	English
Pages (from-to)	469-489
Number of pages	21
Journal	SIAM Journal on Matrix Analysis and Applications
Volume	37
Issue number	1
DOIs	https://doi.org/10.1137/151003143
State	Published - 2016

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Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

Keywords

Bramble-Pasiack cg method
Darcýs law
H(div) preconditioner
Krylov subspace method
Preconditioning technique
Saddle point problems

ASJC Scopus subject areas

Analysis

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10.1137/151003143

Cite this

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title = "Bramble-Pasciak-type conjugate gradient method for Darc{\'y}s equations",

abstract = "We consider the solution of a system of linear algebraic equations which is obtained from a Raviart-Thomas mixed finite element formulation of Darc{\'y}s equations. In [C. E. Powell and D. Silvester, SIAM J. Matrix Anal. Appl., 25 (2003), pp. 718-738], Powell and Silvester developed a block-diagonal preconditioner for this system. In this research work, we extend their results by constructing a block-triangular preconditioner and establish an eigenvalue bound for the preconditioned matrix. The preconditioned matrix is nonsymmetric but it is self-adjoint in a nonstandard inner product. The Bramble-Pasciak-type conjugate gradient method is used to solve the linear system which ensures that the norm of the error is minimized. Numerical experiments confirm the theoretical results and illustrate good convergence properties.",

keywords = "Bramble-Pasiack cg method, Darc{\'y}s law, H(div) preconditioner, Krylov subspace method, Preconditioning technique, Saddle point problems",

author = "Faisal Fairag and Mohammed Alshahrani and Hattan Tawfiq",

note = "Publisher Copyright: Copyright {\textcopyright} by SIAM. Unauthorized reproduction of this article is prohibited.",

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volume = "37",

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AU - Alshahrani, Mohammed

AU - Tawfiq, Hattan

PY - 2016

Y1 - 2016

N2 - We consider the solution of a system of linear algebraic equations which is obtained from a Raviart-Thomas mixed finite element formulation of Darcýs equations. In [C. E. Powell and D. Silvester, SIAM J. Matrix Anal. Appl., 25 (2003), pp. 718-738], Powell and Silvester developed a block-diagonal preconditioner for this system. In this research work, we extend their results by constructing a block-triangular preconditioner and establish an eigenvalue bound for the preconditioned matrix. The preconditioned matrix is nonsymmetric but it is self-adjoint in a nonstandard inner product. The Bramble-Pasciak-type conjugate gradient method is used to solve the linear system which ensures that the norm of the error is minimized. Numerical experiments confirm the theoretical results and illustrate good convergence properties.

AB - We consider the solution of a system of linear algebraic equations which is obtained from a Raviart-Thomas mixed finite element formulation of Darcýs equations. In [C. E. Powell and D. Silvester, SIAM J. Matrix Anal. Appl., 25 (2003), pp. 718-738], Powell and Silvester developed a block-diagonal preconditioner for this system. In this research work, we extend their results by constructing a block-triangular preconditioner and establish an eigenvalue bound for the preconditioned matrix. The preconditioned matrix is nonsymmetric but it is self-adjoint in a nonstandard inner product. The Bramble-Pasciak-type conjugate gradient method is used to solve the linear system which ensures that the norm of the error is minimized. Numerical experiments confirm the theoretical results and illustrate good convergence properties.

KW - Bramble-Pasiack cg method

KW - Darcýs law

KW - H(div) preconditioner

KW - Krylov subspace method

KW - Preconditioning technique

KW - Saddle point problems

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Bramble-Pasciak-type conjugate gradient method for Darcýs equations

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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