Mathematical Modeling and Analysis Parallel, Scalable, and Robust Multigrid on Structured Grids

Introduction: Robust and efficient multilevel iterative solvers are vital for the predictive simulation of complex multiscale and multicomponent nonlinear applications. Specifically, diffusive phenomena play a significant role in wide range of applications, including radiation transport, flow in porous media, and composite materials. In fact, the solution of the diffusive component (elliptic component) of these systems frequently dominates the simulation cost because it is characterized by a discontinuous diffusion coefficient with fine-scale spatial structure. Thus, efficient multilevel iterative methods are crucial because their solution cost scales linearly with the number of unknowns (i.e., optimal algorithmic scaling). In particular, this optimal scaling facilitates the efficient three-dimensional multiscale simulations of linear problems. It also expands the applicability and enhances the effectiveness of large threedimensional multicomponent nonlinear simulations that advance implicitly in time (e.g., matrixfree Newton-Krylov methods) through efficient and robust preconditioning of the Krylov iteration.