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 no 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. In other words, the total iteration count from the preconditioned Krylov subspace method remains fixed regardless of the mesh size and highly varying jump coefficients. Several examples 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

BIO: Dr Fabien is an assistant professor in the department of mathematics at the University of Wisconsin-Madison since 2022, and was previously at Brown University in the Division of Applied Mathematics as Prager assistant professor. In 2019 He received his PhD from Rice University. His research explores numerical methods for partial differential equations, numerical linear algebra, massively parallel computational architectures, and applications to real-world phenomena. Primary areas of focus are discontinuous Galerkin finite element methods, multilevel solvers (multigrid, domain decomposition), and accelerators (GPUs, FPGAs, Xeon Phi, etc.). Example applications include long-wave phenomena (tsunamis, storm surges, ocean waves), subterranean flows (mantle and lava dynamics, groundwater recovery). By leveraging numerical methods, computational science, and computer simulations, my work aims to better understand, predict, and assess complex phenomena that would be otherwise difficult, if not impossible to examine through classical experimentation alone.