  We revisit the problem of spurious modes that are sometimes encountered in
partial differential equations discretizations. It is generally suspected that
one of the causes for spurious modes is due to how boundary conditions are
treated, and we use this as the starting point of our investigations. By
regarding boundary conditions as algebraic constraints on a differential
equation, we point out that any differential equation with homogeneous boundary
conditions also admits a typically infinite number of hidden or implicit
boundary conditions. In most discretization schemes, these additional implicit
boundary conditions are violated, and we argue that this is what leads to the
emergence of spurious modes. These observations motivate two definitions of the
quality of computed eigenvalues based on violations of derivatives of boundary
conditions on the one hand, and on the Grassmann distance between subspaces
associated with computed eigenspaces on the other. Both of these tests are
based on a standardized treatment of boundary conditions and do not require a
priori knowledge of eigenvalue locations. The effectiveness of these tests is
demonstrated on several examples known to have spurious modes. In addition,
these quality tests show that in most problems, about half the computed
spectrum of a differential operator is of low quality. The tests also
specifically identify the low accuracy modes, which can then be projected out
as a type of model reduction scheme.
