Analysis and Computation of Compatible Least-Squares Methods for div-curl Equations
SIAM Journal on Numerical Analysis
Algebraic Multigrid for Linear Systems Obtained by Explicit Element Reduction
SIAM Journal on Scientific Computing
A fast multigrid-based electromagnetic eigensolver for curved metal boundaries on the Yee mesh
Journal of Computational Physics
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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.