Spectra and Pseudospectra of Waveform Relaxation Operators

We describe the spectra and pseudospectra of continuous time and discrete time waveform relaxation operators. Since the spectrum of the finite-interval waveform relaxation operator is only a singleton, the spectrum of the infinite-interval operator is typically used to describe the behavior of waveform relaxation algorithms. Here, we show that the pseudospectrum is a more useful tool for analyzing convergence of waveform iterations on finite intervals and prove that the pseudospectrumof the infinite-interval operator is in fact the limit of the pseudospectrum of the finite-interval operator.