Uzawa Smoother in Multigrid for the Coupled Porous Medium and Stokes Flow System

Luo, Peiyao; Rodrigo, Carmen; Gaspar, Francisco J.; Oosterlee, Cornelis W.

doi:10.1137/16m1076514

Cited by 21 publications

(19 citation statements)

References 42 publications

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“…Thus, it seems natural to assume that this relaxation is also suitable for the coupled Stokes-poroelasticity problem. The smoother is obtained by splitting the discrete operator as follows (25) where M A is a typical smoother for A and ω is some positive parameter. The use of M A makes the approach less costly because of the inexact solve for the velocities and displacements at each iteration.…”

Section: Decoupled Smoother Uzawa Smoothermentioning

confidence: 99%

“…However, the proposed multigrid solution method can be generalized to varying K -values. In [25] (and in [26]), we have performed numerical experiments for the coupled Darcy-Stokes system in which the porous medium was modeled by a random heterogeneous hydraulic conductivity K . When the multigrid algorithm with Uzawa smoother is applied, the optimal relaxation parameter ω is varied within the poroelastic domain, because ω depends on K .…”

Section: Remarkmentioning

confidence: 99%

“…A.16 where 2h high is the space of high frequencies. Details about the basis of LFA can be found in [25]. There are no essential difficulties to obtain bounds for μ A , see for example [30].…”

Section: A2 Discretization Of Stokes Equationsmentioning

confidence: 99%

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Monolithic multigrid method for the coupled Stokes flow and deformable porous medium system

Luo

Rodrigo

Gaspar

et al. 2018

Journal of Computational Physics

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The interaction between fluid flow and a deformable porous medium is a complicated multi-physics problem, which can be described by a coupled model based on the Stokes and poroelastic equations. A monolithic multigrid method together with either a coupled Vanka smoother or a decoupled Uzawa smoother is employed as an efficient numerical technique for the linear discrete system obtained by finite volumes on staggered grids. A specialty in our modeling approach is that at the interface of the fluid and poroelastic medium, two unknowns from the different subsystems are defined at the same grid point. We propose a special discretization at and near the points on the interface, which combines the approximation of the governing equations and the considered interface conditions. In the decoupled Uzawa smoother, Local Fourier Analysis (LFA) helps us to select optimal values of the relaxation parameter appearing. To implement the monolithic multigrid method, grid partitioning is used to deal with the interface updates when communication is required between two subdomains. Numerical experiments show that the proposed numerical method has an excellent convergence rate. The efficiency and robustness of the method are confirmed in numerical experiments with typically small realistic values of the physical coefficients.

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Section: Decoupled Smoother Uzawa Smoothermentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

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Monolithic multigrid method for the coupled Stokes flow and deformable porous medium system

Luo

Rodrigo

Gaspar

et al. 2018

Journal of Computational Physics

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“…Furthermore, we denote by ω an event in the probability space ( , F , P), where is the sample space with σ -field F and probability measure P. Our description of the stochastic flow model follows from [26][27][28] where the deterministic counterpart of this problem is considered.…”

Section: Stochastic Transport In Darcy-stokes Systemmentioning

confidence: 99%

“…A large numerical error or even a reduction in the order of grid convergence may be encountered if interface conditions are not handled properly. A staggered arrangement of the unknowns greatly simplifies the discretization along the interface and has been proven to be effective in reducing numerical error along the interface [28]. Moreover, a staggered grid is also a convenient way of avoiding spurious oscillations in the numerical solution [35] and obtaining conservation of mass throughout the system, also on a relatively coarse grid.…”

Section: Finite Volume Discretizationmentioning

confidence: 99%

A multigrid multilevel Monte Carlo method for transport in the Darcy–Stokes system

Kumar

Luo

Gaspar

et al. 2018

Journal of Computational Physics

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a r t i c l e i n f o a b s t r a c t A multilevel Monte Carlo (MLMC) method for Uncertainty Quantification (UQ) of advectiondominated contaminant transport in a coupled Darcy-Stokes flow system is described.In particular, we focus on high-dimensional epistemic uncertainty due to an unknown permeability field in the Darcy domain that is modelled as a lognormal random field. This paper explores different numerical strategies for the subproblems and suggests an optimal combination for the MLMC estimator. We propose a specific monolithic multigrid algorithm to efficiently solve the steady-state Darcy-Stokes flow with a highly heterogeneous diffusion coefficient. Furthermore, we describe an Alternating Direction Implicit (ADI) based time-stepping for the flux-limited quadratic upwinding discretization for the transport problem. Numerical experiments illustrating the multigrid convergence and cost of the MLMC estimator with respect to the smoothness of permeability field are presented.

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Independence of placement for local Fourier analysis

2021

Numerical Linear Algebra App

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Local Fourier analysis (LFA) serves a prominent role in the prediction of the convergence factor of multigrid methods for discretizations of PDEs. However, few discussions about the implementation of LFA for complicated discretizations of PDEs exist, such as higher order finite elements, as staggered meshes could lead to complex LFA representations of the grid‐transfer operator. In this work, we prove that the LFA representation for d‐dimensional PDEs is independent of the placement of degrees of freedom (DoFs). Intuitively speaking, the seeding of the unknowns has no direct effect on the LFA presentation, instead, it serves as a unitary transformation between different representations. Thus, different LFA representations for a given PDE have the same spectrum and norm. Furthermore, we provide a uniform representation in terms of the location of the unknowns, named simple representation, where we allocate all of the DoFs at nodes, resulting in a simple and unified way to compute the symbols of the discrete operators, especially for the grid‐transfer operators. This simple representation can contribute to the generalization of the implementation of LFA for different types of discretizations and different problems, especially for higher order discretization methods.

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Uzawa Smoother in Multigrid for the Coupled Porous Medium and Stokes Flow System

Cited by 21 publications

References 42 publications

Monolithic multigrid method for the coupled Stokes flow and deformable porous medium system

Monolithic multigrid method for the coupled Stokes flow and deformable porous medium system

A multigrid multilevel Monte Carlo method for transport in the Darcy–Stokes system

Independence of placement for local Fourier analysis

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