Counterexamples to the currently accepted explanation for spurious modes and necessary and sufficient conditions to avoid them

Recently, a new theory concerning the spurious free approximation of electromagnetic eigenproblems has appeared, which is in contrast with the one currently accepted in the electromagnetic community. To stress the need to adopt the new theory, we show that in two simple cases, spurious modes appear, even though the conditions on which the old and accepted theory is based are satisfied. On the contrary, none of these cases conflicts with the new theory. A new and more subdolous behavior of spurious modes is also pointed out.