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Robust Two-Level Overlapping Preconditioner for Solving Darcy Flow in High-Contrast Media


Core Concepts
A robust and efficient two-level overlapping domain decomposition preconditioner is developed for solving linear algebraic systems obtained from simulating Darcy flow in high-contrast media.
Abstract
The key highlights and insights of the content are: The authors develop a two-level overlapping domain decomposition preconditioner for solving linear algebraic systems arising from the discretization of Darcy flow in high-contrast media using the mixed finite element method. The preconditioner utilizes a multiscale coarse space constructed via the Generalized Multiscale Finite Element Method (GMsFEM) to capture the high-contrast features of the permeability field. Local spectral problems are solved in each non-overlapping coarse element to form the coarse space. The local solvers in the preconditioner are derived from additive Schwarz methods with a non-Galerkin discretization to ensure mass conservation of the velocity field. Rigorous analysis is provided to show that the condition number of the preconditioned system can be bounded above under certain assumptions. Extensive numerical experiments with various types of three-dimensional high-contrast models are presented, demonstrating the robustness of the preconditioner against contrast ratios, as well as the influence of parameters such as the number of eigenfunctions, oversampling sizes, and subdomain partitions. Strong and weak scalability performances of the proposed preconditioner are also examined.
Stats
The authors use the following key metrics and figures to support their analysis: The condition number of the preconditioned system can be bounded above as cond(P^-1 A) ≤ C_T / C_R, where C_T and C_R are positive constants. The authors provide rigorous analysis to derive the bounds on C_T and C_R under certain assumptions.
Quotes
"The core idea of these methods is to solve target problems in coarse grids with carefully engineered multiscale basis functions, which are usually solutions of some well-designed sub-problems containing heterogeneity information of the original geological models." "Utilizing eigenfunctions of some carefully designed local spectral problems is to some extent indispensable for constructing a robust and efficient preconditioners."

Deeper Inquiries

How can the proposed preconditioner be extended to handle more complex boundary conditions or coupled flow-transport problems in subsurface modeling

The proposed preconditioner can be extended to handle more complex boundary conditions or coupled flow-transport problems in subsurface modeling by incorporating additional terms in the spectral problems solved in each coarse element. For complex boundary conditions, such as mixed or Robin boundary conditions, the spectral problems can be modified to include the boundary conditions explicitly. This modification would involve adjusting the basis functions and eigenvalue calculations to account for the specific boundary conditions. In the case of coupled flow-transport problems, where both Darcy flow and transport phenomena are involved, the preconditioner can be adapted to consider the interactions between the two processes. This adaptation may require the introduction of additional terms in the spectral problems that capture the coupling effects between flow and transport. By incorporating these additional terms and adjusting the basis functions accordingly, the preconditioner can effectively handle coupled flow-transport problems in subsurface modeling.

What are the potential limitations or drawbacks of the GMsFEM-based approach compared to other multiscale methods, and how can they be addressed

The GMsFEM-based approach, while effective in capturing high-contrast features of the permeability field and providing a robust preconditioner for Darcy flow in high-contrast media, may have some limitations compared to other multiscale methods. One potential limitation is the computational cost associated with solving the spectral problems in each coarse element, especially for large-scale problems with a high number of eigenfunctions. This computational overhead can impact the efficiency of the preconditioner, particularly in terms of scalability. Another limitation could be the reliance on a structured mesh and uniform structured mesh for the fine and coarse elements, respectively. This may restrict the applicability of the method to more general or unstructured meshes commonly encountered in practical subsurface modeling scenarios. To address these limitations, efforts can be made to optimize the spectral problem solvers, explore adaptive mesh strategies, and investigate the use of more flexible basis functions that can accommodate unstructured meshes.

Can the insights from this work on constructing robust preconditioners be applied to other types of PDEs beyond Darcy flow, such as Stokes or Brinkman equations

The insights from this work on constructing robust preconditioners for Darcy flow can be applied to other types of PDEs beyond Darcy flow, such as Stokes or Brinkman equations. The key idea of using multiscale basis functions obtained from solving local spectral problems to construct a coarse space can be generalized to other elliptic or parabolic PDEs. By adapting the spectral problem formulations and basis functions to the specific characteristics of the new PDEs, similar robust preconditioners can be developed. For example, in the case of Stokes equations, which describe the flow of viscous fluids, the spectral problems can be tailored to capture the velocity-pressure coupling inherent in the equations. Similarly, for Brinkman equations, which model flow in porous media with a Darcy flow component and a Stokes flow component, the preconditioner can be designed to address the unique features of these equations. By applying the principles of GMsFEM and leveraging the insights gained from constructing preconditioners for Darcy flow, robust preconditioners can be developed for a wide range of PDEs in subsurface modeling.
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