A non-overlapping domain decomposition method for continuous-pressure mixed finite element approximations of the Stokes problem

Benhassine, Hani; Bendali, Abderrahmane

doi:10.1051/m2an/2010070

Cited by 5 publications

(3 citation statements)

References 34 publications

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“…al. [21], Benhassine and Bendali [1], and Kim and Lee [13]. But the convergence rate analysis of those approaches cannot be applied to the continuous pressure case due to the indefiniteness of the linear systems; such difficulty can often be removed conveniently when discontinuous pressures are used in the discretization.…”

Section: Introductionmentioning

confidence: 99%

A FETI-DP Type Domain Decomposition Algorithm for Three-Dimensional Incompressible Stokes Equations

Tu¹,

Li²

2015

SIAM J. Numer. Anal.

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The FETI-DP (dual-primal finite element tearing and interconnecting) algorithms, proposed by the authors in [SIAM solving incompressible Stokes equations, are extended to three-dimensional problems. A new analysis of the condition number bound for using the Dirichlet preconditioner is given. The algorithm and analysis are valid for mixed finite elements with both continuous and discontinuous pressures. An advantage of this new analysis is that the numerous coarse level velocity components, required in the previous analysis to enforce the divergence-free subdomain boundary velocity conditions, are no longer needed. This greatly reduces the size of the coarse level problem in the algorithm, especially for three-dimensional problems. The coarse level velocity space can be chosen as simple as those coarse spaces for solving scalar elliptic problems corresponding to each velocity component. Both the Dirichlet and lumped preconditioners are analyzed using the same framework in this new analysis. Their condition number bounds are proved to be independent of the number of subdomains for fixed subdomain problem size. Numerical experiments in both two and three dimensions, using mixed finite elements with both continuous and discontinuous pressures, demonstrate the convergence rate of the algorithms.

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Section: Introductionmentioning

confidence: 99%

A FETI-DP Type Domain Decomposition Algorithm for Three-Dimensional Incompressible Stokes Equations

Tu¹,

Li²

2015

SIAM J. Numer. Anal.

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“…al. [26], and by Benhassine and Bendali [1]. In their work, an indefinite system of linear equations need be solved, either by a generalized minimal residual method or simply by a conjugate gradient method.…”

Section: Introductionmentioning

confidence: 99%

A non-overlapping domain decomposition method for incompressible Stokes equations with continuous pressure

Li,

2012

Preprint

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A non-overlapping domain decomposition algorithm is proposed to solve the linear system arising from mixed finite element approximation of incompressible Stokes equations. A continuous finite element space for the pressure is used. In the proposed algorithm, Lagrange multipliers are used to enforce continuity of the velocity component across the subdomain domain boundary. The continuity of the pressure component is enforced in the primal form, i.e., neighboring subdomains share the same pressure degrees of freedom on the subdomain interface and no Lagrange multipliers are needed. After eliminating all velocity variables and the independent subdomain interior parts of the pressures, a symmetric positive semi-definite linear system for the subdomain boundary pressures and the Lagrange multipliers is formed and solved by a preconditioned conjugate gradient method. A lumped preconditioner is studied and the condition number bound of the preconditioned operator is proved to be independent of the number of subdomains for fixed subdomain problem size. Numerical experiments demonstrate the convergence rate of the proposed algorithm.

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“…Previous work on domain decomposition methods for Stokes discretizations with continuous pressure can be found in [8,26,27,41], but without a convergence rate analysis. There has also been earlier work by Pavarino and Scacchi [36] on isogeometric discretizations of Stokes problems, also based on [44], but some theoretical issues remained open and the pressure block of the preconditioner was based on the principal minor of the assembled pressure mass matrix related to the values on the interface.…”

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confidence: 99%

Block FETI–DP/BDDC preconditioners for mixed isogeometric discretizations of three-dimensional almost incompressible elasticity

Widlund¹,

Zampini²,

Scacchi³

et al. 2021

Math. Comp.

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A block FETI-DP/BDDC preconditioner for mixed formulations of almost incompressible elasticity is constructed and analyzed; FETI-DP (dual primal finite element tearing and interconnection) and BDDC (balancing domain decomposition by constraints) are two very successful domain decomposition algorithms for a variety of elliptic problems. The saddle point problems of the mixed problems are discretized with mixed isogeometric analysis with continuous pressure fields. As in previous work by Tu and Li (2015), for finite element discretizations of the incompressible Stokes system, the proposed preconditioner is applied to a reduced positive definite system involving only the pressure interface variable and the Lagrange multipliers of the FETI-DP algorithm. The novelty of this preconditioner consists in using BDDC with deluxe scaling for the interface pressure block as well as deluxe scaling for the FETI-DP preconditioner for the Lagrange multiplier block. A convergence rate analysis is presented with a condition number bound for the preconditioned operator which depends on the inf-sup parameter of the fully assembled problem and the condition number of a closely related BDDC algorithm for compressible elasticity. This bound is scalable in the number of subdomains, poly-logarithmic in the ratio of subdomain and element sizes, and robust with respect to material incompressibility and presence of discontinuities of the Lamé parameters across subdomain interfaces. Parallel numerical experiments validate the theory and indicate how the rate of convergence varies with respect to the spline polynomial degree and regularity and the deformation of the domain. Of particular interest is the development of variants of the algorithm with a coarse component of small dimension.

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A non-overlapping domain decomposition method for continuous-pressure mixed finite element approximations of the Stokes problem

Cited by 5 publications

References 34 publications

A FETI-DP Type Domain Decomposition Algorithm for Three-Dimensional Incompressible Stokes Equations

A FETI-DP Type Domain Decomposition Algorithm for Three-Dimensional Incompressible Stokes Equations

A non-overlapping domain decomposition method for incompressible Stokes equations with continuous pressure

Block FETI–DP/BDDC preconditioners for mixed isogeometric discretizations of three-dimensional almost incompressible elasticity

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