SIAM Journal on Scientific Computing
This paper presents a multigrid method for solving variable coefficient Maxwell's equations. The novelty in this method is the use of interpolation operators that do not produce multilevel commutativity complexes that lead to multilevel exactness. Rather, the effects of multilevel exactness are built into the level equations themselves-on the finest level using a discrete T -V formulation and on the coarser grids through the Galerkin coarsening procedure of a T - V formulation. These built-in structures permit the levelwise use of an effective hybrid smoother on the curl-free near-nullspace components and permit the development of interpolation operators for handling the curl-free and divergence-free error components separately. The resulting block-diagonal interpolation operator does not satisfy multilevel commutativity but has good approximation properties for both of these error components. Applying operator-dependent interpolation for each of these error components leads to an effective multigrid scheme for variable coefficient Maxwell's equations, where multilevel commutativity-based methods can degrade. Numerical results are presented to verify the effectiveness of this new scheme.
Jones, J., & Lee, B. (2006). A multigrid method for variable coefficient maxwell's equations. SIAM Journal on Scientific Computing, 27(5), 1689-1708.