Efficient Monolithic Two-Level Schwarz Preconditioner for Biot's Consolidation Model in Two Space Dimensions

Core Concepts

A monolithic two-level Schwarz preconditioner is constructed and analyzed for the efficient iterative solution of the quasi-static Biot problem in three-field formulation, discretized using strongly mass-conserving mixed finite element methods.

Abstract

The paper addresses the construction and analysis of a class of domain decomposition methods for the iterative solution of the quasi-static Biot problem in three-field formulation. The considered discrete model arises from time discretization by the implicit Euler method and space discretization by a family of strongly mass-conserving methods exploiting Hdiv-conforming approximations of the solid displacement and fluid flux fields.
For the resulting saddle-point problem, the authors construct monolithic overlapping domain decomposition (DD) methods whose analysis relies on a transformation into an equivalent symmetric positive definite system and on stable decompositions of the involved finite element spaces under proper problem-dependent norms.
The key steps are:

Transformation of the saddle-point problem into an equivalent, singularly perturbed symmetric positive definite system.
Derivation of stable decompositions of the displacement and velocity spaces into divergence-free and curl-free subspaces.
Proof of uniform convergence of the additive and multiplicative two-level Schwarz methods based on the stable decompositions.

Numerical results on two-dimensional test problems are in accordance with the provided theoretical uniform convergence estimates for the two-level multiplicative Schwarz method.

Stats

The permeability tensor K is assumed to satisfy two-sided bounds, i.e., there exist positive constants Rmin and Rmax such that
Rmin ||v||^2_Ω ≤ (K^-1 v, v)_Ω ≤ Rmax ||v||^2_Ω for all v in [L^2(Ω)]^2.

Quotes

"The key task in the development of such methods is the stability analysis of the underlying saddle-point problems, which can be performed in the framework presented in [HKLP23]."
"By a decomposition into physical subsystems, the coupled static problem can be solved implicitly, using a loose or explicit coupling, or an iterative coupling. The latter, often provides an attractive alternative in terms of achieving high accuracy at reasonable computational cost."

Key Insights Distilled From

Monolithic two-level Schwarz preconditioner for Biot's consolidation model in two space dimensions

by Stefan Megge... at arxiv.org 04-26-2024

https://arxiv.org/pdf/2404.16684.pdf

Deeper Inquiries

How can the presented analysis be extended to handle high contrast in the permeability tensor or strongly anisotropic materials

To extend the analysis to handle high contrast in the permeability tensor or strongly anisotropic materials, adjustments need to be made in the stability analysis and preconditioning strategies.

High Contrast Permeability Tensor:

For high contrast permeability tensors, the bounds on the permeability tensor need to be carefully considered.
The stability analysis should account for a wider range of values for the permeability tensor, ensuring that the preconditioner remains effective across different scales.
Techniques such as weighted Poincaré inequalities can be employed to handle the variability in the permeability tensor.

Strongly Anisotropic Materials:

In the case of strongly anisotropic materials, the preconditioner design should take into account the directional variations in material properties.
Anisotropy can affect the convergence behavior, so the preconditioner should be robust in handling such variations.
The analysis should include considerations for the impact of anisotropy on the stability and convergence of the iterative solver.

By incorporating these considerations into the analysis and preconditioning techniques, the method can be extended to effectively handle high contrast permeability tensors and strongly anisotropic materials.

What are the potential advantages and limitations of the monolithic approach compared to partitioned or iterative coupling schemes for solving the Biot problem

Advantages of Monolithic Approach:

Global Information: The monolithic approach considers the entire system simultaneously, allowing for the incorporation of global information and interactions between fields.
Consistency: It ensures consistency in the solution process, avoiding errors that may arise from separate treatments of different fields.
Efficiency: By solving the coupled system in a unified manner, the monolithic approach can potentially reduce computational costs and memory requirements.
Limitations of Monolithic Approach:

Complexity: Handling the entire system at once can lead to increased complexity in implementation and analysis.
Scalability: The monolithic approach may face challenges in scalability for large systems due to the need to solve a larger, more complex linear system.
Robustness: The method's robustness may be influenced by the choice of discretization, preconditioning, and solver parameters.
Comparison with Partitioned or Iterative Coupling Schemes:

Advantages: The monolithic approach avoids issues related to data transfer between subdomains, potentially leading to better convergence rates and accuracy.
Limitations: It may be computationally more expensive than partitioned schemes, especially for large systems. Additionally, it may require more memory and computational resources.

Can the proposed two-level Schwarz preconditioner be combined with other techniques, such as multigrid or algebraic multigrid, to further improve the efficiency and robustness of the overall solver

The proposed two-level Schwarz preconditioner can be effectively combined with other techniques to enhance the efficiency and robustness of the overall solver:

Multigrid Methods: Integrating the two-level Schwarz preconditioner with multigrid methods can lead to faster convergence rates by efficiently smoothing out errors at different scales. The combination can improve the overall solver performance, especially for large-scale problems.

Algebraic Multigrid: By incorporating algebraic multigrid techniques, the solver can adapt to the problem's characteristics without relying on the geometric information. This can lead to improved convergence and efficiency, particularly for complex geometries and highly heterogeneous materials.

Domain Decomposition Methods: Combining the two-level Schwarz preconditioner with domain decomposition methods can further enhance parallelism and scalability. By distributing the computational load across multiple subdomains, the solver can achieve better performance on parallel architectures.

By leveraging these complementary techniques in conjunction with the two-level Schwarz preconditioner, the solver can achieve superior efficiency, scalability, and robustness in solving the Biot problem.