A frequently used class of simple non-self-adjoint elliptic PDE test problems is shown to result on discretization in matrices having highly specialized spectra, to the extent that such test problems may well not be representative of realproblems. A tentative suggestion is made of a slightly more complicated class of test problems, which it is hoped would avoid the difficulty. An appendix examines some of the properties of the spectra of finite-difference matrices from non-self-adjoint problems. © 1981 1981, by Academic Press Inc. (London) Limited.