A Novel Algebraic Multigrid-Based Approach for Maxwell’s Equations

This paper presents a new algebraic multigrid-based method for solving the curl-curl formulation of Maxwell’s equations discretized with edge elements. The ultimate goal of this approach is two-fold. The first is to produce a multiple-coarsening multigrid method with two approximately decoupled hierarchies branching off at the initial coarse level, one resolving the divergence-free error and the other resolving the curl-free error, i.e., a multigrid method that couples only on the finest level and mimics a Helmholtz decomposition on the coarse levels. The second consideration is to produce the hierarchies using a non-agglomerate coarsening scheme. To roughly attain this two-fold goal, this new approach constructs the first coarse level using topological properties of the mesh. In particular, a discrete orthogonal decomposition of the finest edges is constructed by dividing them into two sets, those forming a spanning tree and the complement set forming the cotree. Since the cotree edges do not form closed cycles, these edges cannot support “complete” nearnullspace gradient functions of the curl-curl Maxwell operator. Thus, partitioning the finest level matrix using this tree/cotree decomposition, the cotree-cotree submatrix does not have a large nearnullspace. Hence, a non-agglomerate algebraic multigrid method (AMG) that can handle strong positive and negative off-diagonal elements can be applied to this submatrix. This cotree operator is related to the initial coarse-grid operator for the divergence-free hierarchy. The curl-free hierarchy is generated by a nodal Poisson operator obtained by restricting the Maxwell operator to the space of gradients. Unfortunately, because the cotree operator itself is not the initial coarse-grid operator for the divergence-free hierarchy, the multiple-coarsening scheme composed of the cotree matrix and its coarsening, and the nodal Poisson operator and its coarsening does not give an overall efficient method. Algebraically, the tree/cotree coupling on the finest level, which is accentuated through smooth divergence-free error, is too strong to be handled sufficiently only on the finest level. In this new approach, these couplings are handled using oblique/orthogonal projections onto the space of discretely divergence-free vectors. In the multigrid viewpoint, the initial coarsening from the target fine level to the divergence-free subspace is obtained using these oblique/orthogonal restriction/interpolation operators in the Galerkin coarsening procedure. The resulting coarse grid operator can be preconditioned with a product operator involving a cotree-cotree submatrix and a topological matrix related to a discrete Poisson operator. The overall iteration is then a multigrid cycle for a nodal Poisson operator (the curl-free branch) coupled on the finest grid to a preconditioned Krylov iteration for the fine grid Maxwell operator restricted to the subspace of discretely divergencefree vectors. Numerical results are presented to verify the effectiveness and difficulties of this new approach for solving the curl-curl formulation of Maxwell’s equations.