An Efficient Multiscale Preconditioner for Large-Scale Highly Heterogeneous Flow

We propose an efficient and robust multiscale preconditioner for large-scale incompressible flow in highly heterogeneous porous media. We start from the discretization of the first-order form for the single phase incompressible flow and apply a velocity elimination strategy to obtain an equation with pressure as the only unknown. Then an efficient preconditioner is designed to solve this equation. The key component of the preconditioner is the adoption of a nonstandard coarse space that contains the media’s heterogeneity information. We solve a carefully constructed spectral problem in each coarse element to form the nonstandard coarse space. Then a rigorous convergence analysis for the proposed two-grid algorithm is carried out where the key ingredients lie in the smoothing property of the ILU(0) smoother and the approximation property. In particular, our analysis shows that our preconditioner is robust and efficient thanks to this newly constructed coarse space. Rich numerical tests with several types of large-scale, three-dimensional, highly heterogeneous permeability fields with resolutions up to 1024³ are presented. The experimental results show that our generalized multiscale space–based preconditioner is robust with respect to the contrast, size, and geometry of the permeability fields. Moreover, the complexity of the preconditioner is close to O⁡(Dof). We also successfully apply this preconditioner for multiphase flow simulation and transport problems arising from reservoir simulation. © 2023 Society for Industrial and Applied Mathematics.