Multigrid methods based on shifted inverse iteration for the Maxwell eigenvalue problem
Provides theoretically grounded multigrid solvers for a challenging eigenvalue problem in computational electromagnetics, but the methods are incremental extensions of existing techniques.
The paper develops two multigrid methods (Rayleigh quotient iteration and inverse iteration with fixed shift) for the Maxwell eigenvalue problem with discontinuous coefficients, proving uniform convergence and asymptotically optimal error estimates, confirmed by numerical experiments.
In this paper two types of multgrid methods, i.e., the Rayleigh quotient iteration and the inverse iteration with fixed shift, are developed for solving the Maxwell eigenvalue problem with discontinuous relative magnetic permeability and electric permittivity. With the aid of the mixed form of source problem associated with the eigenvalue problem, we prove the uniform convergence of the discrete solution operator to the solution operator in $L^2(Ω)$ using discrete compactness of edge element space. Then we prove the asymptotically optimal error estimates for both multigrid methods. Numerical experiments confirm our theoretical analysis.