A scalable block-preconditioning strategy for divergence-conforming B-spline discretizations of the Stokes problem

A. M.A. Côrtes; L. Dalcin; A. F. Sarmiento; N. Collier; V. M. Calo

doi:10.1016/j.cma.2016.10.014

A scalable block-preconditioning strategy for divergence-conforming B-spline discretizations of the Stokes problem

A. M.A. Côrtes, L. Dalcin, A. F. Sarmiento, N. Collier, V. M. Calo

Computational Earth Sciences Group

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

8 Scopus citations

Abstract

The recently introduced divergence-conforming B-spline discretizations allow the construction of smooth discrete velocity–pressure pairs for viscous incompressible flows that are at the same time inf–sup stable and pointwise divergence-free. When applied to the discretized Stokes problem, these spaces generate a symmetric and indefinite saddle-point linear system. The iterative method of choice to solve such system is the Generalized Minimum Residual Method. This method lacks robustness, and one remedy is to use preconditioners. For linear systems of saddle-point type, a large family of preconditioners can be obtained by using a block factorization of the system. In this paper, we show how the nesting of “black-box” solvers and preconditioners can be put together in a block triangular strategy to build a scalable block preconditioner for the Stokes system discretized by divergence-conforming B-splines. Besides the known cavity flow problem, we used for benchmark flows defined on complex geometries: an eccentric annulus and hollow torus of an eccentric annular cross-section.

Original language	English
Pages (from-to)	839-858
Number of pages	20
Journal	Computer Methods in Applied Mechanics and Engineering
Volume	316
DOIs	https://doi.org/10.1016/j.cma.2016.10.014
State	Published - Apr 1 2017

Bibliographical note

Publisher Copyright:
© 2016 Elsevier B.V.

Funding

This publication was made possible in part by a National Priorities Research Program grant 7-1482-1-278 from the Qatar National Research Fund (a member of The Qatar Foundation), by the European Union's Horizon 2020 Research and Innovation Program under the Marie Sklodowska-Curie grant agreement No. 644602 and the Center for Numerical Porous Media at King Abdullah University of Science and Technology (KAUST). The J. Tinsley Oden Faculty Fellowship Research Program at the Institute for Computational Engineering and Sciences (ICES) of the University of Texas at Austin has partially supported the visits of VMC to ICES. The authors acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing HPC resources that have contributed to the research results reported within this paper.

Funders	Funder number
European Union's Horizon 2020 research and innovation program
Institute for Computational Engineering and Sciences
Marie Sklodowska-Curie	644602
Texas Advanced Computing Center
University of Texas at Austin
Qatar National Research Fund
Horizon 2020 Framework Programme	644202
King Abdullah University of Science and Technology

Keywords

B-spline compatible vector field discretization
Block preconditioners
Isogeometric analysis
Krylov subspace methods
Stokes flow

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10.1016/j.cma.2016.10.014

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title = "A scalable block-preconditioning strategy for divergence-conforming B-spline discretizations of the Stokes problem",

abstract = "The recently introduced divergence-conforming B-spline discretizations allow the construction of smooth discrete velocity–pressure pairs for viscous incompressible flows that are at the same time inf–sup stable and pointwise divergence-free. When applied to the discretized Stokes problem, these spaces generate a symmetric and indefinite saddle-point linear system. The iterative method of choice to solve such system is the Generalized Minimum Residual Method. This method lacks robustness, and one remedy is to use preconditioners. For linear systems of saddle-point type, a large family of preconditioners can be obtained by using a block factorization of the system. In this paper, we show how the nesting of “black-box” solvers and preconditioners can be put together in a block triangular strategy to build a scalable block preconditioner for the Stokes system discretized by divergence-conforming B-splines. Besides the known cavity flow problem, we used for benchmark flows defined on complex geometries: an eccentric annulus and hollow torus of an eccentric annular cross-section.",

keywords = "B-spline compatible vector field discretization, Block preconditioners, Isogeometric analysis, Krylov subspace methods, Stokes flow",

author = "C{\^o}rtes, {A. M.A.} and L. Dalcin and Sarmiento, {A. F.} and N. Collier and Calo, {V. M.}",

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AU - Côrtes, A. M.A.

AU - Dalcin, L.

AU - Sarmiento, A. F.

AU - Collier, N.

AU - Calo, V. M.

PY - 2017/4/1

Y1 - 2017/4/1

N2 - The recently introduced divergence-conforming B-spline discretizations allow the construction of smooth discrete velocity–pressure pairs for viscous incompressible flows that are at the same time inf–sup stable and pointwise divergence-free. When applied to the discretized Stokes problem, these spaces generate a symmetric and indefinite saddle-point linear system. The iterative method of choice to solve such system is the Generalized Minimum Residual Method. This method lacks robustness, and one remedy is to use preconditioners. For linear systems of saddle-point type, a large family of preconditioners can be obtained by using a block factorization of the system. In this paper, we show how the nesting of “black-box” solvers and preconditioners can be put together in a block triangular strategy to build a scalable block preconditioner for the Stokes system discretized by divergence-conforming B-splines. Besides the known cavity flow problem, we used for benchmark flows defined on complex geometries: an eccentric annulus and hollow torus of an eccentric annular cross-section.

AB - The recently introduced divergence-conforming B-spline discretizations allow the construction of smooth discrete velocity–pressure pairs for viscous incompressible flows that are at the same time inf–sup stable and pointwise divergence-free. When applied to the discretized Stokes problem, these spaces generate a symmetric and indefinite saddle-point linear system. The iterative method of choice to solve such system is the Generalized Minimum Residual Method. This method lacks robustness, and one remedy is to use preconditioners. For linear systems of saddle-point type, a large family of preconditioners can be obtained by using a block factorization of the system. In this paper, we show how the nesting of “black-box” solvers and preconditioners can be put together in a block triangular strategy to build a scalable block preconditioner for the Stokes system discretized by divergence-conforming B-splines. Besides the known cavity flow problem, we used for benchmark flows defined on complex geometries: an eccentric annulus and hollow torus of an eccentric annular cross-section.

KW - B-spline compatible vector field discretization

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KW - Isogeometric analysis

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A scalable block-preconditioning strategy for divergence-conforming B-spline discretizations of the Stokes problem

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Keywords

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