A highly parallelized multiscale preconditioner for Darcy flow in high-contrast media

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: $W_L\subset W_{L-1}\subset \dots \subset W_1=W_h$. By defining an appropriate spectral problem within the space of $W_{i-1}$, we leverage the eigenfunctions of these spectral problems to form $W_i$. 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.