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

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. Lastly, 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.
Authors:
 [1] ;  [2] ;  [3] ; ORCiD logo [4] ;  [5]
  1. King Abdullah Univ. of Science and Technology (KAUST), Thuwal (Saudi Arabia). Center for Numerical Porous Media (NumPor); Federal Univ. of the State of Rio de Janeiro (UNIRIO), Rio de Janeiro (Brazil)
  2. King Abdullah Univ. of Science and Technology (KAUST), Thuwal (Saudi Arabia). Center for Numerical Porous Media (NumPor); Consejo Nacional de Investigaciones Cientificas y Tecnicas (CONICET), Santa Fe (Argentina). Centro de Investigacion de Metodos Computacionales (CIMEC)
  3. King Abdullah Univ. of Science and Technology (KAUST), Thuwal (Saudi Arabia). Center for Numerical Porous Media (NumPor); King Abdullah Univ. of Science and Technology (KAUST), Thuwal (Saudi Arabia). Applied Mathematics and Computational Science (AMCS)
  4. Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Computer Science and Mathematics Division
  5. King Abdullah Univ. of Science and Technology (KAUST), Thuwal (Saudi Arabia). Center for Numerical Porous Media (NumPor); King Abdullah Univ. of Science and Technology (KAUST), Thuwal (Saudi Arabia). Applied Mathematics and Computational Science (AMCS); King Abdullah Univ. of Science and Technology (KAUST), Thuwal (Saudi Arabia). Earth Science and Engineering (ErSE); Curtin Univ., Perth, WA (Australia). Western Australian School of Mines; Commonwealth Scientific and Industrial Research Organisation (CSIRO), Kensington, WA (Australia). Mineral Resources
Publication Date:
Grant/Contract Number:
AC05-00OR22725; 7-1482-1-278; 644602
Type:
Accepted Manuscript
Journal Name:
Computer Methods in Applied Mechanics and Engineering
Additional Journal Information:
Journal Volume: 316; Journal ID: ISSN 0045-7825
Publisher:
Elsevier
Research Org:
Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
Sponsoring Org:
USDOE; Qatar Foundation; European Union (EU); King Abdullah University of Science and Technology (KAUST)
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Isogeometric analysis; B-spline compatible vector field discretization; Krylov subspace methods; Block preconditioners; Stokes flow
OSTI Identifier:
1399434

Cortes, Adriano M., Dalcin, Lisandro, Sarmiento, Adel F., Collier, Nathaniel O., and Calo, Victor M.. A scalable block-preconditioning strategy for divergence-conforming B-spline discretizations of the Stokes problem. United States: N. p., Web. doi:10.1016/j.cma.2016.10.014.
Cortes, Adriano M., Dalcin, Lisandro, Sarmiento, Adel F., Collier, Nathaniel O., & Calo, Victor M.. A scalable block-preconditioning strategy for divergence-conforming B-spline discretizations of the Stokes problem. United States. doi:10.1016/j.cma.2016.10.014.
Cortes, Adriano M., Dalcin, Lisandro, Sarmiento, Adel F., Collier, Nathaniel O., and Calo, Victor M.. 2016. "A scalable block-preconditioning strategy for divergence-conforming B-spline discretizations of the Stokes problem". United States. doi:10.1016/j.cma.2016.10.014. https://www.osti.gov/servlets/purl/1399434.
@article{osti_1399434,
title = {A scalable block-preconditioning strategy for divergence-conforming B-spline discretizations of the Stokes problem},
author = {Cortes, Adriano M. and Dalcin, Lisandro and Sarmiento, Adel F. and Collier, Nathaniel O. and Calo, Victor M.},
abstractNote = {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. Lastly, 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.},
doi = {10.1016/j.cma.2016.10.014},
journal = {Computer Methods in Applied Mechanics and Engineering},
number = ,
volume = 316,
place = {United States},
year = {2016},
month = {10}
}