p-Multilevel Preconditioners for HHO Discretizations of the Stokes Equations with Static Condensation

doi:10.1007/s42967-021-00142-5

Communications on Applied Mathematics and Computation ›› 2022, Vol. 4 ›› Issue (3): 783-822.doi: 10.1007/s42967-021-00142-5

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p-Multilevel Preconditioners for HHO Discretizations of the Stokes Equations with Static Condensation

Lorenzo Botti¹, Daniele A. Di Pietro²

1. Department of Engineering and Applied Sciences, University of Bergamo, Bergamo, Italy;
2. IMAG, Univ Montpellier, CNRS, Montpellier, France

收稿日期:2020-09-28 修回日期:2021-04-21 出版日期:2022-09-20 发布日期:2022-07-04
通讯作者: Lorenzo Botti,E-mail:lorenzo.botti@unibg.it;Daniele A. Di Pietro,E-mail:daniele.di-pietro@umontpellier.fr E-mail:lorenzo.botti@unibg.it;daniele.di-pietro@umontpellier.fr
基金资助:
Daniele Di Pietro acknowledges the support of Agence Nationale de la Recherche Grant fast4hho (ANR-17-CE23-0019).

p-Multilevel Preconditioners for HHO Discretizations of the Stokes Equations with Static Condensation

Lorenzo Botti¹, Daniele A. Di Pietro²

1. Department of Engineering and Applied Sciences, University of Bergamo, Bergamo, Italy;
2. IMAG, Univ Montpellier, CNRS, Montpellier, France

Received:2020-09-28 Revised:2021-04-21 Online:2022-09-20 Published:2022-07-04
Contact: Lorenzo Botti,E-mail:lorenzo.botti@unibg.it;Daniele A. Di Pietro,E-mail:daniele.di-pietro@umontpellier.fr E-mail:lorenzo.botti@unibg.it;daniele.di-pietro@umontpellier.fr
Supported by:
Daniele Di Pietro acknowledges the support of Agence Nationale de la Recherche Grant fast4hho (ANR-17-CE23-0019).

摘要/Abstract

摘要： We propose a p-multilevel preconditioner for hybrid high-order (HHO) discretizations of the Stokes equation, numerically assess its performance on two variants of the method, and compare with a classical discontinuous Galerkin scheme. An efficient implementation is proposed where coarse level operators are inherited using $ L^2 $-orthogonal projections defined over mesh faces and the restriction of the fine grid operators is performed recursively and matrix-free. Both h- and k-dependency are investigated tackling two- and three-dimensional problems on standard meshes and graded meshes. For the two HHO formulations, featuring discontinuous or hybrid pressure, we study how the combination of p-coarsening and static condensation influences the V-cycle iteration. In particular, two different static condensation procedures are considered for the discontinuous pressure HHO variant, resulting in global linear systems with a different number of unknowns and matrix non-zero entries. Interestingly, we show that the efficiency of the solution strategy might be impacted by static condensation options in the case of graded meshes.

关键词: Stokes equations, Divergence free constraint, Hybrid high-order, Discontinuous Galerkin, p-multigrid, Static condensation

Abstract: We propose a p-multilevel preconditioner for hybrid high-order (HHO) discretizations of the Stokes equation, numerically assess its performance on two variants of the method, and compare with a classical discontinuous Galerkin scheme. An efficient implementation is proposed where coarse level operators are inherited using $ L^2 $-orthogonal projections defined over mesh faces and the restriction of the fine grid operators is performed recursively and matrix-free. Both h- and k-dependency are investigated tackling two- and three-dimensional problems on standard meshes and graded meshes. For the two HHO formulations, featuring discontinuous or hybrid pressure, we study how the combination of p-coarsening and static condensation influences the V-cycle iteration. In particular, two different static condensation procedures are considered for the discontinuous pressure HHO variant, resulting in global linear systems with a different number of unknowns and matrix non-zero entries. Interestingly, we show that the efficiency of the solution strategy might be impacted by static condensation options in the case of graded meshes.

Key words: Stokes equations, Divergence free constraint, Hybrid high-order, Discontinuous Galerkin, p-multigrid, Static condensation

中图分类号:

Lorenzo Botti, Daniele A. Di Pietro. p-Multilevel Preconditioners for HHO Discretizations of the Stokes Equations with Static Condensation[J]. Communications on Applied Mathematics and Computation, 2022, 4(3): 783-822.

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p-Multilevel Preconditioners for HHO Discretizations of the Stokes Equations with Static Condensation

p-Multilevel Preconditioners for HHO Discretizations of the Stokes Equations with Static Condensation

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