Cartesian PML approximation to resonances in open systems in R2R2

In this paper, we consider a Cartesian PML approximation to resonance values of time-harmonic problems posed on unbounded domains in R2R2. A PML is a fictitious layer designed to find solutions arising from wave propagation and scattering problems supplemented with an outgoing radiation condition at infinity. Solutions obtained by a PML coincide with original solutions near wave sources or scatterers while they decay exponentially as they propagate into the layer. Due to rapid decay of solutions, it is natural to truncate unbounded domains to finite regions of computational interest. In this analysis, we introduce a PML in Cartesian geometry to transform a resonance problem (characterized as an eigenvalue problem with improper eigenfunctions) on an unbounded domain to a standard eigenvalue problem on a finite computational region. Truncating unbounded domains gives rise to perturbation of resonance values, however we show that eigenvalues obtained by the truncated problem converge to resonance values as the size of computational domain increases. In addition, our analysis shows that this technique is free of spurious resonance values provided truncated domains are sufficiently large. Finally, we present the results of numerical experiments with simple model problems.