An Algebraic Multigrid Approach Based on a Compatible Gauge Reformulation of Maxwell's Equations

With the rise in popularity of compatible finite element, finite difference, and finite volume discretizations for the time domain eddy current equations, there has been a corresponding need for fast solvers of the resulting linear algebraic systems. However, the traits that make compatible discretizations a preferred choice for the Maxwell's equations also render these linear systems essentially intractable by truly black-box techniques. We propose an algebraic reformulation of the discrete eddy current equations along with a new algebraic multigrid (AMG) technique for this reformulated problem. The reformulation process takes advantage of a discrete Hodge decomposition on cochains to replace the discrete eddy current equations by an equivalent $2\times2$ block linear system whose diagonal blocks are discrete Hodge-Laplace operators acting on 1-cochains and 0-cochains, respectively. While this new AMG technique requires somewhat specialized treatment on the finest mesh, the coarser meshes can be handled using standard methods for Laplace-type problems. Our new AMG method is applicable to a wide range of compatible methods on structured and unstructured grids, including edge finite elements, mimetic finite differences, covolume methods, and Yee-like schemes. We illustrate the new technique, using edge elements in the context of smoothed aggregation AMG, and present computational results for problems in both two and three dimensions.