With the rising popularity of compatible discretizations (edge elements) for the eddy current Maxwell's equations, there is a corresponding need for fast solvers. We propose an algebraic reformulation of the discrete system along with a new AMG technique for this reformulated problem. This transforms the curl-curl system into a block 2x2 system where the diagonal blocks are an edge Hodge Laplacian and a nodal scalar Laplacian, respectively.

In this talk, we describe some basic properties of the de Rham complex and the Hodge Laplacian. Using these properties, we define a transformation that can be applied to the linear system associated with an edge element discretization of the eddy current equations. The key point is that the resulting linear system is much more ameneable to algebraic mutlgrid (AMG) and yields an identical discrete solution to the original eddy current system. In particular, an AMG method must now be applied to two Laplace-type systems (one a vector Hodge Laplacian and the other a scalar nodal Laplacian). A second talk will discuss the AMG method and illustrate the computational efficiency of the reformulation.