Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A robust two-level overlapping preconditioner for Darcy flow in high-contrast media

Changqing Ye, Shubin Fu, Eric T. Chung, Jizu Huang

Abstract

In this article, a two-level overlapping domain decomposition preconditioner is developed for solving linear algebraic systems obtained from simulating Darcy flow in high-contrast media. Our preconditioner starts at a mixed finite element method for discretizing the partial differential equation by Darcy's law with the no-flux boundary condition and is then followed by a velocity elimination technique to yield a linear algebraic system with only unknowns of pressure. Then, our main objective is to design a robust and efficient domain decomposition preconditioner for this system, which is accomplished by engineering a multiscale coarse space that is capable of characterizing high-contrast features of the permeability field. A generalized eigenvalue problem is solved in each non-overlapping coarse element in a communication-free manner to form the global solver, which is accompanied by local solvers originated from additive Schwarz methods but with a non-Galerkin discretization to derive the two-level preconditioner. We provide a rigorous analysis that indicates that the condition number of the preconditioned system could be bounded above with several assumptions. Extensive numerical experiments with various types of three-dimensional high-contrast models are exhibited. In particular, we study the robustness against the contrast of the media as well as the influences of numbers of eigenfunctions, oversampling sizes, and subdomain partitions on the efficiency of the proposed preconditioner. Besides, strong and weak scalability performances are also examined.

A robust two-level overlapping preconditioner for Darcy flow in high-contrast media

Abstract

In this article, a two-level overlapping domain decomposition preconditioner is developed for solving linear algebraic systems obtained from simulating Darcy flow in high-contrast media. Our preconditioner starts at a mixed finite element method for discretizing the partial differential equation by Darcy's law with the no-flux boundary condition and is then followed by a velocity elimination technique to yield a linear algebraic system with only unknowns of pressure. Then, our main objective is to design a robust and efficient domain decomposition preconditioner for this system, which is accomplished by engineering a multiscale coarse space that is capable of characterizing high-contrast features of the permeability field. A generalized eigenvalue problem is solved in each non-overlapping coarse element in a communication-free manner to form the global solver, which is accompanied by local solvers originated from additive Schwarz methods but with a non-Galerkin discretization to derive the two-level preconditioner. We provide a rigorous analysis that indicates that the condition number of the preconditioned system could be bounded above with several assumptions. Extensive numerical experiments with various types of three-dimensional high-contrast models are exhibited. In particular, we study the robustness against the contrast of the media as well as the influences of numbers of eigenfunctions, oversampling sizes, and subdomain partitions on the efficiency of the proposed preconditioner. Besides, strong and weak scalability performances are also examined.
Paper Structure (14 sections, 7 theorems, 57 equations, 8 figures, 4 tables, 1 algorithm)

This paper contains 14 sections, 7 theorems, 57 equations, 8 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The model problem and the mixed finite element discretization
  4. The velocity elimination technique
  5. Two-level overlapping preconditioners
  6. GMsFEM based two-level overlapping preconditioner
  7. Analysis
  8. Assumption A
  9. Assumption B
  10. Numerical experiments
  11. Contrast robustness
  12. Scalability
  13. Parameter tests
  14. Conclusion

Key Result

Lemma 4.1

\newlabellem:fsl0 Let $\mathsf{b}(\cdot,\cdot)$ be a symmetric, positive semi-definite bilinear form on $\underline{V}$ with a matrix representation as $\mathcal{R}:\underline{V} \rightarrow \mathbb{R}^n$ be a linear map, and $\mathcal{R}^*:\mathbb{R}^n\rightarrow \underline{V}$ be an adjoint operator by Assume that $\mathcal{R}$ maps $\ker(\underline{\mathsf{B}})$ into $\ker(\mathsf{A})$, and

Figures (8)

  • Figure 1: An illustration of the two-scale mesh, a fine element $\tau$, a coarse element $K_i$ and its oversampling coarse element $K_i^m$ with $m=2$.
  • Figure 1: (left column) the 3- and 5-channel configurations; (middle column) the $11$ smallest eigenvalues calculated by setting $\tilde{\kappa}=\kappa$ w.r.t. different $\kappa^*$ that is the value of $\kappa$ in channels, and the top (bottom) plot is corresponding to the 3-channel (5-channel) configuration; (right column) the $11$ smallest eigenvalues calculated by setting $\tilde{\kappa}=1$ w.r.t. different $\kappa^*$, and the top (bottom) plot is corresponding to the 3-channel (5-channel) configuration.
  • Figure 1: An illustration for the proof of \ref{['lem:A_i']}, where $\mathtt{k}_{p,q}$, $\mathtt{k}_{p-1,q}$, $\mathtt{k}_{p+1,q}$, $\mathtt{k}_{p,q-1}$ and $\mathtt{k}_{p,q+1}$ are values of $\kappa$ on the fine elements respectively, the oversampling coarse element $K_i^m$ with $m=2$ is filled with gray and also decorated with dashed borderlines, and the fine element in the top-left corner of $K_i^m$ is highlighted.
  • Figure 1: (left part) the distributions of $\alpha$ for different configurations in a periodic cell, (right part) The number of GMRES iterations $\mathtt{iter}$ against $L_\star$ under different $\mathtt{cr}$.
  • Figure 2: An illustration for a construction of $\tilde{\kappa}$, where $\mathtt{k}_{p,q}$, $\mathtt{k}_{p-1,q}$, $\mathtt{k}_{p+1,q}$, $\mathtt{k}_{p,q-1}$ and $\mathtt{k}_{p,q+1}$ are values of $\kappa$ on the located fine elements respectively, the fine element in the top-left corner of $K_i$ is highlighted with four edges $e_1\sim e_4$ surrounded, $\mathtt{v}_{p,q}$, $\mathtt{v}_{p+1,q}$ and $\mathtt{v}_{p,q-1}$ are values of $v\in W_h(K_i)$ on the located fine elements respectively.
  • ...and 3 more figures

Theorems & Definitions (14)

  • Remark 3.1
  • Lemma 4.1
  • Proof 1
  • Lemma 4.2
  • Proof 2
  • Remark 4.3
  • Lemma 4.4
  • Proof 3
  • Lemma 4.5
  • Lemma 4.6
  • ...and 4 more