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An efficient multiscale multigrid preconditioner for Darcy flow in high-contrast media

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

Abstract

In this paper, we develop a multigrid preconditioner to solve Darcy flow in highly heterogeneous porous media. The key component of the preconditioner is to construct a sequence of nested subspaces $W_{\mathcal{L}}\subset W_{\mathcal{L}-1}\subset\cdots\subset W_1=W_h$. An appropriate spectral problem is defined in the space of $W_{i-1}$, then the eigenfunctions of the spectral problems are utilized to form $W_i$. The preconditioner is applied to solve a positive semidefinite linear system which results from discretizing the Darcy flow equation with the lowest order Raviart-Thomas spaces and adopting a trapezoidal quadrature rule. Theoretical analysis and numerical investigations of this preconditioner will be presented. In particular, we will consider several typical highly heterogeneous permeability fields whose resolutions are up to $1024^3$ and examine the computational performance of the preconditioner in several aspects, such as strong scalability, weak scalability, and robustness against the contrast of the media. We also demonstrate an application of this preconditioner for solving a two-phase flow benchmark problem.

An efficient multiscale multigrid preconditioner for Darcy flow in high-contrast media

Abstract

In this paper, we develop a multigrid preconditioner to solve Darcy flow in highly heterogeneous porous media. The key component of the preconditioner is to construct a sequence of nested subspaces . An appropriate spectral problem is defined in the space of , then the eigenfunctions of the spectral problems are utilized to form . The preconditioner is applied to solve a positive semidefinite linear system which results from discretizing the Darcy flow equation with the lowest order Raviart-Thomas spaces and adopting a trapezoidal quadrature rule. Theoretical analysis and numerical investigations of this preconditioner will be presented. In particular, we will consider several typical highly heterogeneous permeability fields whose resolutions are up to and examine the computational performance of the preconditioner in several aspects, such as strong scalability, weak scalability, and robustness against the contrast of the media. We also demonstrate an application of this preconditioner for solving a two-phase flow benchmark problem.
Paper Structure (13 sections, 3 theorems, 54 equations, 8 figures, 4 tables, 1 algorithm)

This paper contains 13 sections, 3 theorems, 54 equations, 8 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The model problem and the discretization
  4. The velocity elimination technique
  5. Multiscale coarse space based multigrid preconditioner
  6. Analysis
  7. Numerical experiments
  8. Scalability tests
  9. Robustness tests on a fractured medium
  10. Tests on the second strategy
  11. Comparison with the exact preconditioner
  12. Application on two-phase flow problem
  13. Conclusion

Key Result

Lemma 4.1

Let $C_\star$ be a positive constant such that $\widetilde{\mathsf{M}}\lesssim C_\star\mathsf{A}$, $C_\lambda$ be $\min\{*\}{\lambda_{l_\mathup{c}^{i}}}_{i=1}^{m_\mathup{c}}$. Then, the bound holds, where $C_\mathup{s}$ is a generic positive constant.

Figures (8)

  • Figure 1: An illustration of the expression of $\kappa_e$. Depending on the direction, $\kappa_{i,j+1/2}$ and $\kappa_{i+1/2,j}$ utilize harmonic averages of different component of the permeability field.
  • Figure 2: An illustration of hierarchical meshes, a fine element $\tau$, a coarse element $K_\mathup{c}$, a coarse-coarse element $K_\mathup{cc}$ and ghost layers. In MPI-implementations, each $K_\mathup{cc}$ is assigned to a unique process.
  • Figure 3: (a) The periodic cell configuration used for the scalability tests. (b) An illustration of a domain that consists of $4\times 4\times 4$ cells, where long channels across the domain. (c)-left the results of strong scalability tests, where the DoF is fixed as $512^3$ and the number of MPI processes varies from $\{4^3, 5^3, 6^3, 7^3, 8^3\}$. (c)-right the results of weak scalability tests, where the ratio of $\mathtt{DoF}$ and $\mathtt{proc}$ is fixed around $85^3$.
  • Figure 4: A fractured medium model, where red and blue regions are referred to as matrix (place $1$ as the conductivity) and fractures (place $10^\mathtt{cr}$ as the conductivity) respectively.
  • Figure 5: Rock properties of the SPE10 model: (a) $\kappa^x$ or $\kappa^y$, (b) $\kappa^z$, (c) $\kappa^x\!/\kappa^z$, and (d) modified $\phi$.
  • ...and 3 more figures

Theorems & Definitions (3)

  • Lemma 4.1
  • Theorem 4.2
  • Theorem 4.3