Maxwell's Equations: Discrete Reformulation and Algebraic Multigrid Solution

Ray Tuminaro
Sandia National Laboratories

ABSTRACT: We consider the linear solution of the eddy current equations. The large null space of the curl-curl operator within the eddy current equations complicates the application of most standard preconditioning techniques. Most current solvers are specialized techniques that cannot effectively leverage standard algebraic multigrid (AMG) methods and software for Laplace-type problems. We propose a new completely algebraic reformulation of the discrete eddy current equations along with a new AMG algorithm for this reformulated problem. The reformulation process takes advantage of a discrete Hodge decomposition to replace the discrete eddy current equations by an equivalent block 2x2 linear system whose diagonal blocks are discrete Hodge Laplace operators acting on edges and nodes, respectively. While this new AMG technique requires some special treatment in generating a grid transfer from the fine 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, co-volume 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.