A geometric multigrid method for isogeometric compatible discretizations of the generalized Stokes and Oseen problems

Coley, Christopher; Benzaken, Joseph; Evans, John A.

doi:10.1002/nla.2145

Cited by 13 publications

(13 citation statements)

References 52 publications

(121 reference statements)

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“…We extend the iterated penalty method, as in [57], to perform penalty iterations in tandem with nonlinear iterations. It is also possible to use the multigrid method proposed in [60] to solve the saddle point problem.…”

Section: Implementation Using Fenics and Tigarmentioning

confidence: 99%

Variational multiscale modeling with discretely divergence-free subscales

Evans

Kamensky

Bazilevs

2020

Computers & Mathematics with Applications

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We introduce a residual-based stabilized formulation for incompressible Navier-Stokes flow that maintains discrete (and, for divergence-conforming methods, strong) mass conservation for inf-sup stable spaces with H 1 -conforming pressure approximation, while providing optimal convergence in the diffusive regime, robustness in the advective regime, and energetic stability. The method is formally derived using the variational multiscale (VMS) concept, but with a discrete fine-scale pressure field which is solved for alongside the coarse-scale unknowns such that the coarse and fine scale velocities separately satisfy discrete mass conservation. We show energetic stability for the full Navier-Stokes problem, and we prove convergence and robustness for a linearized model (Oseen flow), under the assumption of a divergence-conforming discretization. Numerical results indicate that all properties extend to the fully nonlinear case and that the proposed formulation can serve to model unresolved turbulence.

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Section: Implementation Using Fenics and Tigarmentioning

confidence: 99%

Variational multiscale modeling with discretely divergence-free subscales

Evans

Kamensky

Bazilevs

2020

Computers & Mathematics with Applications

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“…Isogeometric preconditioners for the Stokes system have also been studied in recent papers: [21,22] consider block-diagonal and block-triangular preconditioners combined to black-box solvers (either algebraicmultigrid or incomplete factorization); [23] studies the domain-decomposition FETI-DP strategy; [24] focuses on a multigrid strategy; another multigrid approach, which extends the results of [19], can be found in [25].…”

Section: Introductionmentioning

confidence: 99%

Robust isogeometric preconditioners for the Stokes system based on the Fast Diagonalization method

Montardini

Sangalli

Tani

2018

Computer Methods in Applied Mechanics and Engineering

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In this paper we propose a new class of preconditioners for the isogeometric discretization of the Stokes system. Their application involves the solution of a Sylvester-like equation, which can be done efficiently thanks to the Fast Diagonalization method. These preconditioners are robust with respect to both the spline degree and mesh size. By incorporating information on the geometry parametrization and equation coefficients, we maintain efficiency on non-trivial computational domains and for variable kinematic viscosity. In our numerical tests we compare to a standard approach, showing that the overall iterative solver based on our preconditioners is significantly faster.In conclusion our numerical benchmarks confirm that the proposed preconditioner is very efficient and well suited for the k-method. Further advances in the solver performance can be achieved with a matrix-free approach, that accelerates the matrix-vector multiplication operation, for moderate or large degree. A first step in this research direction is [30].The outline of the paper is as follows. In Section 2 we give a short review of the Taylor-Hood and Raviart-Thomas isogeometric discretizations for the Stokes system, and summarize the main properties of the Kronecker product. The derivation of the discrete Stokes system is given in Section 3, while in Section 4 we introduce some standard block-structured preconditioners that we will consider in the numerical tests. The core of the paper is Section 5, where we focus on the construction of the preconditioning matrices for the velocity and pressure blocks, discuss their properties and solution strategies. In the Section 6 we propose the modification aimed at improving the preconditioner efficiency by incorporating some information on the geometry parametrization. Numerical results on three different single-patch domains are reported in Section 7. Finally, in Section 8 we draw the conclusions and discuss future directions of research. Preliminaries B-splinesIn this section we summarize some basic concepts of B-spline based isogeometric analysis, referring to [2] for the details.Given m and p two positive integers, we introduce a knot vector Ξ := {0 = ξ 1 ≤ ... ≤ ξ m+p+1 = 1} and the associated breakpoint vector Z := {ζ 1 , ..., ζ s }, which contains knots without repetitions. We use open knot vectors, i.e. we suppose ξ 1 = ... = ξ p+1 = 0 and ξ m = ... = ξ m+p+1 = 1 .Then, according to Cox-De Boor recursion formulas [31], we define univariate B-splines as: for p = 0:b 0 α,i (η) =

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“…While symmetric positive definite problems cover a large variety of problems in computational mechanics, the developed preconditioning technique is not immediately applicable to nonsymmetric and mixed formulations in, in particular, flow problems. However, multigrid methods are commonly applied in mesh-fitting flow problems, see e.g., [60], and have been observed to be effective with very similar Schwarz (or Vanka [79]) blocks in [71]. Furthermore, it is demonstrated in [50] that Schwarz-type methods can effectively resolve the cut-element-specific conditioning problems in immersed flow problems.…”

Section: Resultsmentioning

confidence: 99%

“…This contribution only considers SPD problems. Multigrid methods are an established concept in fluid mechanics as well [60], however, and similar Schwarz-type methods have successfully been applied to flow problems with both immersed finite element methods [50] and mesh-fitting multigrid solvers [71], i.e., Vanka-smoothers [79]. Therefore, it is anticipated that the presented preconditioning technique extends matatis mutandis to non-SPD and mixed formulations.…”

Section: Introductionmentioning

confidence: 99%

Multigrid solvers for immersed finite element methods and immersed isogeometric analysis

et al. 2019

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Ill-conditioning of the system matrix is a well-known complication in immersed finite element methods and trimmed isogeometric analysis. Elements with small intersections with the physical domain yield problematic eigenvalues in the system matrix, which generally degrades efficiency and robustness of iterative solvers. In this contribution we investigate the spectral properties of immersed finite element systems treated by Schwarz-type methods, to establish the suitability of these as smoothers in a multigrid method. Based on this investigation we develop a geometric multigrid preconditioner for immersed finite element methods, which provides mesh-independent and cut-element-independent convergence rates. This preconditioning technique is applicable to higher-order discretizations, and enables solving large-scale immersed systems in parallel, at a computational cost that scales linearly with the number of degrees of freedom. The performance of the preconditioner is demonstrated for conventional Lagrange basis functions and for isogeometric discretizations with both uniform B-splines and locally refined approximations based on truncated hierarchical B-splines.

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A geometric multigrid method for isogeometric compatible discretizations of the generalized Stokes and Oseen problems

Cited by 13 publications

References 52 publications

Variational multiscale modeling with discretely divergence-free subscales

Variational multiscale modeling with discretely divergence-free subscales

Robust isogeometric preconditioners for the Stokes system based on the Fast Diagonalization method

Multigrid solvers for immersed finite element methods and immersed isogeometric analysis

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