Stability of perfectly matched layers for Maxwell's equations in rectangular solids

Abstract

Perfectly matched layers are extensively used to compute approximate solutions for Maxwell's equations in using a bounded computational domain, usually a rectangular solid. A smaller rectangular domain of interest is surrounded by layers designed to absorb outgoing waves in perfectly reflectionless manner. On the boundary of the computational domain, an absorbing boundary condition is imposed that is necessarily imperfect. The method replaces the Maxwell equations by a larger system, and introduces absorption coefficients positive in the layers. Well posedness of the resulting initial boundary value problem is proved here for the first time. The Laplace transform of a resulting Helmholtz system is studied. For positive real values of the transform variable , the Helmholtz system has a unique solution from a variational form that yields limited regularity for rectangular domains. When is not real the complex variational form loses positivity. We smooth the domain and, in spite of this loss, construct solutions with uniform estimates. Using the regularity, we deduce Maxwell from Helmholtz, then remove the smoothing. The boundary condition at the smoothed boundary must be carefully chosen. A method of Jerison‐Kenig‐Mitrea is extended to compensate the nonpositivity of the flux.