Neumann–Dirichlet maps and analysis of spectral pollution for non-self-adjoint elliptic PDEs with real essential spectrum

We prove that one of the most commonly used techniques for approximating the spectra of certain classes of non-self-adjoint elliptic partial differential equations on exterior domains does not suffer from spectral pollution except possibly in the spectral gaps. This generalizes a well-known result from the self-adjoint case. We also show how the method can be used in conjunction with some simple tricks to avoid spectral pollution for the self-adjoint case. Our proofs are based on a new approach to the nesting set analysis for Neumann to Dirichlet maps first proposed by Amrein and Pearson in 2004, with enhanced convergence results obtained from an elliptic regularity bootstrapping procedure. The numerical results in Section 6 illustrate a technique for finding eigenvalues in spectral gaps of self-adjoint operators by means of a compactly supported complex shift. This method seems to be of independent interest and can be understood without reading the rest of the paper.