In this paper, we develop a highly parallelized preconditioner based on multiscale space to tackle Darcy flow in highly heterogeneous porous media. The crucial component of this preconditioner is devising a sequence of nested subspaces: WL⊂WL−1⊂…⊂W1=Wh. By defining an appropriate spectral problem within the space of Wi−1, we leverage the eigenfunctions of these spectral problems to form Wi. The preconditioner is then employed to solve a positive semidefinite linear system, which arises from discretizing the Darcy flow equation using the lowest order Raviart-Thomas spaces and adopting a trapezoidal quadrature rule. We will present both theoretical analysis and numerical investigations of this preconditioner. In particular, we will explore various highly heterogeneous permeability fields with resolutions of up to 10243, evaluating the computational performance of the preconditioner in several aspects, including strong scalability, weak scalability, and robustness against the contrast ratio of the media. In high-contrast settings, the proposed preconditioner demonstrates superior performance in terms of stability and efficiency compared to the default algebraic multigrid solver in PETSc, a renowned high performance computing library. A numerical experiment will showcase the preconditioner's capability to solve a high-contrast, large-scale problem with 10243 degrees of freedom using just 1728 CPU cores with 30 seconds. Furthermore, we will demonstrate the application of this preconditioner in solving benchmark problems related to two-phase flow.

Publication:

JOURNAL OF COMPUTATIONAL PHYSICS

http://dx.doi.org/10.1016/j.jcp.2024.113603

Author:

Changqing Ye

Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong Special Administrative Region

Shubin Fu

Eastern Institute for Advanced Study, Ningbo, 315200, Zhejiang, PR China

Corresponding author

E-mail address: sfu@eitech.edu.cn

Eric T. Chung

Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong Special Administrative Region

Jizu Huang

LSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, PR China d School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, PR China