Analysis on block diagonal and triangular preconditioners for a PML system of an electromagnetic scattering problem

We shall propose several block triangular preconditioners for a PML system of an electromagnetic wave scattering problem and analyze the spectral behavior of the preconditioned systems. When the PML system is discretized by edge element methods, it results in a discrete system with its stiffness matrix being complex, symmetric but indefinite, which can be formulated into a real symmetric but indefinite saddle-point system. In order to preserve the symmetry of the coefficient matrix, we present block triangular preconditioners with two-sided preconditioning for the discrete PML system. We will estimate the lower and upper bounds of positive and negative eigenvalues of the preconditioned matrices, respectively. On the other hand, one may also like to apply some iteration methods for nonsymmetric linear systems in applications although the discrete systems are symmetric. To this end, we propose a block triangular preconditioner to precondition the systems only from one side and analyze the spectrum of the preconditioned systems. In addition, we have also established a spectral estimate of the preconditioned system by an effective preconditioner that was recently developed in literature. Numerical experiments are presented to demonstrate the effectiveness and robustness of these new preconditioners and our theoretical predictions on the spectral bounds of the preconditioned systems.