An optimal iterative solver for symmetric indefinite systems stemming from mixed approximation

David J. Silvester; Valeria Simoncini

doi:10.1145/1916461.1916466

An optimal iterative solver for symmetric indefinite systems stemming from mixed approximation

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Abstract

We discuss the design and implementation of a suite of functions for solving symmetric indefinite linear systems associated with mixed approximation of systems of PDEs. The novel feature of our iterative solver is the incorporation of error control in the natural "energy" norm in combination with an a posteriori estimator for the PDE approximation error. This leads to a robust and optimally efficient stopping criterion: the iteration is terminated as soon as the algebraic error is insignificant compared to the approximation error. We describe a "proof of concept" MATLAB implementation of this algorithm, which we call EST-MINRES, and we illustrate its effectiveness when integrated into the Incompressible Flow Iterative Solution Software (IFISS) package (cf. ACM Transactions on Mathematical Software 33, Article 14, 2007). © 2011 ACM.

Original language	English
Article number	42
Journal	ACM Transactions on Mathematical Software
Volume	37
Issue number	4
DOIs	https://doi.org/10.1145/1916461.1916466
Publication status	Published - Feb 2011

Keywords

EST-MINRES
Finite elements
Incompressible flow
Iterative solvers
MATLAB
Stopping criteria

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10.1145/1916461.1916466

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abstract = "We discuss the design and implementation of a suite of functions for solving symmetric indefinite linear systems associated with mixed approximation of systems of PDEs. The novel feature of our iterative solver is the incorporation of error control in the natural {"}energy{"} norm in combination with an a posteriori estimator for the PDE approximation error. This leads to a robust and optimally efficient stopping criterion: the iteration is terminated as soon as the algebraic error is insignificant compared to the approximation error. We describe a {"}proof of concept{"} MATLAB implementation of this algorithm, which we call EST-MINRES, and we illustrate its effectiveness when integrated into the Incompressible Flow Iterative Solution Software (IFISS) package (cf. ACM Transactions on Mathematical Software 33, Article 14, 2007). {\textcopyright} 2011 ACM.",

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AB - We discuss the design and implementation of a suite of functions for solving symmetric indefinite linear systems associated with mixed approximation of systems of PDEs. The novel feature of our iterative solver is the incorporation of error control in the natural "energy" norm in combination with an a posteriori estimator for the PDE approximation error. This leads to a robust and optimally efficient stopping criterion: the iteration is terminated as soon as the algebraic error is insignificant compared to the approximation error. We describe a "proof of concept" MATLAB implementation of this algorithm, which we call EST-MINRES, and we illustrate its effectiveness when integrated into the Incompressible Flow Iterative Solution Software (IFISS) package (cf. ACM Transactions on Mathematical Software 33, Article 14, 2007). © 2011 ACM.

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KW - Finite elements

KW - Incompressible flow

KW - Iterative solvers

KW - MATLAB

KW - Stopping criteria

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JO - ACM Transactions on Mathematical Software

JF - ACM Transactions on Mathematical Software

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ER -

An optimal iterative solver for symmetric indefinite systems stemming from mixed approximation

Abstract

Keywords

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