insight - Numerical Methods - # Two-level overlapping Schwarz preconditioners for high-order elliptic problems

Core Concepts

The paper proposes a novel universal construction of two-level overlapping Schwarz preconditioners for 2mth-order elliptic boundary value problems, where m is a positive integer. The coarse space construction can be applied to any finite element discretization for any m that satisfies some common assumptions, ensuring scalable convergence rates.

Abstract

The paper introduces two-level overlapping Schwarz preconditioners for solving 2mth-order elliptic boundary value problems, where m is a positive integer. The key aspect is the construction of a universal coarse space that can be applied to any finite element discretization for any m, without depending on the specific discretization method.
The main highlights are:
The coarse space is constructed using conforming finite element spaces defined on a coarse mesh, with a universal approach that is independent of both m and the type of finite element discretization.
The proposed preconditioners ensure uniformly bounded condition numbers of the preconditioned operators when the ratio of subdomain size to overlap is fixed, regardless of the problem size.
Numerical results are presented for fourth- and sixth-order problems discretized using various conforming, nonconforming, and discontinuous Galerkin-type finite elements, demonstrating the scalability of the proposed preconditioners.

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by Jongho Park at **arxiv.org** 03-29-2024

Deeper Inquiries

The proposed universal coarse space construction can be extended to handle more general elliptic problems by adapting the construction method to accommodate variable coefficients or nonlinear terms. For problems with variable coefficients, the basis functions in the coarse space can be adjusted to reflect the spatial variations in the coefficients. This may involve incorporating additional information about the coefficient functions into the construction of the coarse basis functions. For problems with nonlinear terms, the construction of the coarse space may need to consider the nonlinearity in the problem and ensure that the coarse basis functions can capture the behavior of the nonlinear terms. By appropriately modifying the construction of the coarse space to account for these variations, the universal coarse space approach can be extended to handle a wider range of elliptic problems.

The ideas presented in this work can indeed be applied to develop scalable solvers for other high-order partial differential equations, such as the Cahn-Hilliard or Kohn-Sham equations. By adapting the universal coarse space construction and the two-level overlapping Schwarz preconditioning technique to the specific discretizations and formulations of these equations, efficient solvers can be developed. For the Cahn-Hilliard equation, which describes phase separation in materials, the high-order finite element discretization can benefit from the universal coarse space construction to improve scalability and convergence rates. Similarly, for the Kohn-Sham equations in density functional theory, the high-order discretization can be preconditioned using the two-level overlapping Schwarz method to enhance the efficiency of solving these quantum mechanical problems.

The efficient high-order solvers developed in this work have numerous potential applications in various fields, including material science, optimal control, and phase-field modeling. In material science, these solvers can be used to simulate complex material behaviors with high accuracy and efficiency, enabling researchers to study material properties, phase transitions, and material interactions at a finer level of detail. In optimal control applications, the high-order solvers can be employed to solve optimization problems with high-dimensional state spaces and complex constraints, allowing for more precise control strategies and improved performance. In phase-field modeling, the solvers can facilitate the simulation of phase transitions, crystal growth, and other phenomena with high spatial resolution and accuracy, leading to advancements in materials design and process optimization.

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