Effect of choices of boundary conditions on the numerical efficiency of direct solutions of finite difference frequency domain systems with perfectly matched layers.

Abstract

Direct solvers are a common method for solving finite difference frequency domain (FDFD) systems that arise in numerical solutions of Maxwell's equations. In a direct solver, one factorizes the system matrix. Since the system matrix is typically very sparse, the fill-in of these factors is the single most important computational consideration in terms of time complexity and memory requirements. As a result, it is of great interest to determine ways in which this fill-in can be systematically reduced. In this paper, we show that in the context of commonly used perfectly matched boundary layer methods, the choice of boundary condition behind the perfectly matched boundary layer can be exploited to reduce fill-in incurred during the factorization, leading to significant gains of up to 40% in the efficiency of the factorization procedure. We illustrate our findings by solving linear systems and eigenvalue problems associated with the FDFD method.