Efficient Preconditioners for Coupled Stokes-Darcy Flow Problems

Core Concepts

Robust and efficient preconditioners are developed and analyzed for the coupled Stokes-Darcy flow problems to accelerate the convergence of the applied Krylov method.

Abstract

The content discusses the development and analysis of preconditioners for coupled Stokes-Darcy flow problems. The key highlights are:

The coupled Stokes-Darcy system is discretized using the finite volume method on staggered grids (MAC scheme), which yields large, sparse, ill-conditioned, and non-symmetric linear systems.

Three main classes of preconditioners are considered: block diagonal, block triangular, and constraint preconditioners. Exact and efficient inexact variants of these preconditioners are developed.

Spectral and field-of-values (FOV) analysis is provided to show that the eigenvalues of the preconditioned matrices are clustered and bounded away from zero, ensuring fast convergence of the iterative Krylov method (GMRES).

The proposed preconditioners are shown to be robust with respect to the choice of physical parameters and grid width, making them suitable for large-scale simulations.

Numerical experiments demonstrate the effectiveness and robustness of the developed preconditioners for both the Beavers-Joseph and Beavers-Joseph-Saffman interface conditions.

Stats

The discretization of the coupled Stokes-Darcy problem yields large, sparse, ill-conditioned, and non-symmetric linear systems.
The entries of the coupling matrix C are of order O(h), where h is the grid width.

Quotes

"Robust and efficient preconditioners are needed to accelerate convergence of the applied Krylov method."
"FOV theory states that the convergence of the GMRES method is then independent of the grid width."
"The proposed preconditioners are shown to be robust with respect to the choice of physical parameters and grid width, making them suitable for large-scale simulations."

Key Insights Distilled From

Efficient preconditioners for coupled Stokes-Darcy problems

by Paula Strohb... at arxiv.org 04-30-2024

https://arxiv.org/pdf/2404.18639.pdf

Deeper Inquiries

How can the developed preconditioners be extended to handle more complex coupled flow problems, such as those involving nonlinear effects or multiphase flows

The developed preconditioners can be extended to handle more complex coupled flow problems by incorporating additional terms or modifications to account for nonlinear effects or multiphase flows. For nonlinear effects, the preconditioners can be adapted to include terms that capture the nonlinearity of the equations, such as introducing nonlinear operators or iterative strategies within the preconditioning process. This can help improve the convergence of the iterative solvers when dealing with nonlinearities in the coupled flow equations.
In the case of multiphase flows, the preconditioners can be enhanced to consider the interactions between different phases, such as incorporating additional interface conditions or phase-specific properties into the preconditioning strategy. This can help address the challenges associated with multiphase flow simulations, such as phase transitions, saturation changes, and phase interactions at the fluid interfaces.
By extending the preconditioners to handle these more complex scenarios, the efficiency and robustness of the iterative solvers for coupled flow problems can be improved, enabling more accurate and reliable simulations in diverse applications.

What are the potential limitations of the finite volume discretization approach used in this work, and how could alternative discretization methods impact the design and performance of the preconditioners

The potential limitations of the finite volume discretization approach used in this work include issues related to numerical diffusion, grid orientation effects, and accuracy in capturing complex flow phenomena. Finite volume methods are known to introduce numerical diffusion, especially in high-gradient regions, which can impact the accuracy of the solutions, particularly for problems with sharp interfaces or boundary layers.
Additionally, the finite volume discretization approach may struggle with capturing complex flow behaviors, such as vortices, turbulence, or multiphase interactions, due to its inherent averaging nature over control volumes. This can limit the ability to accurately model intricate flow physics and phenomena in the system.
Alternative discretization methods, such as finite element methods or spectral methods, could potentially offer advantages in terms of accuracy, flexibility in mesh generation, and ability to capture complex geometries or flow behaviors. These alternative methods may impact the design and performance of the preconditioners by requiring adaptations to the matrix structures, preconditioning strategies, or convergence criteria to effectively address the characteristics of the new discretization scheme.

Can the insights gained from the spectral and FOV analysis of the preconditioned systems be leveraged to develop adaptive preconditioning strategies that dynamically adjust to changes in the problem parameters or discretization

The insights gained from the spectral and field-of-values (FOV) analysis of the preconditioned systems can be leveraged to develop adaptive preconditioning strategies that dynamically adjust to changes in the problem parameters or discretization. By understanding the spectral properties of the preconditioned matrices, adaptive preconditioners can be designed to automatically tune their parameters or structures based on the specific characteristics of the problem at hand.
For example, adaptive preconditioners can adjust the level of approximation or the type of preconditioning technique based on the eigenvalue distribution of the preconditioned matrix, ensuring optimal convergence rates for different problem configurations. By monitoring the FOV bounds and spectral clustering patterns, adaptive strategies can dynamically switch between preconditioning methods to maintain efficiency and robustness in varying simulation scenarios.
Overall, leveraging the spectral and FOV analysis to develop adaptive preconditioning strategies can enhance the performance and versatility of iterative solvers for coupled flow problems, enabling more efficient and reliable simulations across a range of applications.