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Sparsifying preconditioner for the time-harmonic Maxwell’s equations. (English) Zbl 1416.78038
Summary: This paper presents the sparsifying preconditioner for the time-harmonic Maxwell’s equations in the integral formulation. Following the work on sparsifying preconditioner for the Lippmann-Schwinger equation [the second author, Multiscale Model. Simul. 13, No. 2, 644–660 (2015; Zbl 1317.65087)], this paper generalizes that approach from the scalar wave case to the vector case. The key idea is to construct a sparse approximation to the dense system by minimizing the non-local interactions in the integral equation, which allows for applying sparse linear solvers to reduce the computational cost. When combined with the standard GMRES solver, the number of preconditioned iterations remains small and essentially independent of the frequency. This suggests that, when the sparsifying preconditioner is adopted, solving the dense integral system can be done as efficiently as solving the sparse system from PDE discretization.
##### MSC:
 78M25 Numerical methods in optics (MSC2010) 65F08 Preconditioners for iterative methods 65F50 Computational methods for sparse matrices 78A45 Diffraction, scattering 78M15 Boundary element methods applied to problems in optics and electromagnetic theory
FDFD
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