A study of the condition number of various finite element matrices involved in the numerical solution of Maxwell's equations

We consider the solution of the time-harmonic Maxwell's equations inside a bounded domain on the boundary of which various conditions are prescribed, including a perfectly matched layer (PML) termination. This problem arises when, e.g., the electromagnetic fields scattered from an inhomogeneous penetrable structure are computed by using a hybrid finite element (FE) and integral equation method in conjunction with a domain decomposition technique. In each of the subdomains, the discretization process leads to a linear system, and an iterative solver may be advantageously utilized when the number of unknowns is large. In this case, the number of iterations and, hence, the computational time required to achieve a given numerical accuracy are known to increase with the condition number κ of the FE matrix. In this paper, we attempt to draw the rules that govern the behavior of κ. To this effect, an eigenmodes technique is proposed that allows to dissociate the influence of the FE mesh and FE basis functions from the one of the actual physical cavity. Numerical examples are provided for one- and three-dimensional problems that illustrate the results so obtained.