Toward an h-Independent Algebraic Multigrid Method for Maxwell's Equations

Abstract

We propose a new algebraic multigrid (AMG) method for solving the eddy current approximations to Maxwell's equations. This AMG method has its roots in an algorithm proposed by Reitzinger and Schöberl. The main focus in the Reitzinger and Schöberl method is to maintain null-space properties of the weak ∇ × ∇× operator on coarse grids. While these null-space properties are critical, they are not enough to guarantee h-independent convergence rates of the overall multigrid scheme. We present a new strategy for choosing intergrid transfers that not only maintains the important null-space properties on coarse grids but also yields significantly improved multigrid convergence rates. This improvement is related to those we explored in a previous paper, but is fundamentally simpler, easier to compute, and performs better with respect to both multigrid operator complexity and convergence rates. The new strategy builds on ideas in smoothed aggregation to improve the approximation property of an existing interpolation operator. By carefully choosing the smoothing operators, we show how it is sometimes possible to achieve h-independent convergence rates with a modest increase in multigrid operator complexity. Though this ideal case is not always possible, the overall algorithm performs significantly better than the original scheme in both iterations and run time. Finally, the Reitzinger and Schöberl method, as well as our previous smoothed method, are shown to be special cases of this new algorithm.

Block Iterative Methods and Recycling for Improved Scalability of Linear Solvers

“…Because we use complex-valued scalars, hypre, and in particular its Maxwell solver AMS [72], cannot be used. MueLu [73] Trilinos can only solve Maxwell's equation in eddy current formulation 6 , and it is not clear how it handles high-order edge elements. Figure 7a is a plot of the time to solution, including both the setup and the solution phases, for solving the linear system on 512 up to 4,096 subdomains.…”

ALEGRA-HEDP : version 4.6.

“…The Aztec [46] and ML [44] documentation give a descriptions of options. The three-dimensional magnetic field discretization and multigrid solution research and development can be reviewed in several references [5,6,7,21,39].…”

ALEGRA-MHD : version 4.6

“…The Aztec [28] and ML [27] documentation give a descriptions of options. The three-dimensional magnetic field discretization and multigrid solution research and development can be reviewed in several references [1,2,3,12,24].…”