Seminars & Events

CSE Community Seminar | October 17, 2025

Presenter Maurice Fabien, MLK Visiting Professor, MIT; Assistant professor of mathematics, University of Wisconsin-Madison

Talk Title Robust multilevel solvers for porous media applications

Abstract

Common models for heterogeneous flow in porous media (hydrology applications, industrial and environmental processes), such as the Darcy equation or Brinkman equation, are partial differential equations (PDEs) which can have highly varying nonlinear jump coefficients. For accurate approximations, this requires a careful choice of numerical discretization. In this talk, I will present standard discontinuous Galerkin (DG) discretizations to approximate the Darcy and Brinkman equations. The DG discretization gives rise to a large sparse linear system of equations which has to be resolved. The conditioning of the linear system is very poor due to the highly varying jump coefficients and the higher-order nature of the DG method.

Multilevel techniques (e.g., multigrid, domain decomposition) are well known to be among the most efficient linear solvers for the discretizations that arise from elliptic PDEs. Many multilevel preconditioners assume that there is a meaningful geometric mesh hierarchy available, which is not the case in many applications (e.g., direct flow simulation using high-resolution micro-computed tomographic ($\mu$-CT) images of porous rock). Straightforward application of preexisting multilevel solvers to non-conforming, totally discontinuous discretizations tend to result in methods that do not scale. This is especially noticeable in the presence of highly varying jump coefficients on unstructured meshes, and for positive semidefinite linear systems.
      
A novel multilevel preconditioned Krylov subspace is introduced. The preconditioner is provably optimal in the sense that the condition number remains bounded independent of the contrast in material parameters, as well as the mesh size. Several examples at the pore scale and darcy scale will demonstrate the robustness of the preconditioner. Some noteworthy features of the preconditioner are: the highly heterogeneous permeability does not need to be aligned with the mesh, the heterogeneous permeability can be anisotropic, and no geometric hierarchy is required for the computational domain.