Use of the finite-element method (FEM) for the computation of transient thermal problems can lead to the anomalous result that, even in the absence of local sources of heat, temperatures near suddenly cooled boundaries show an initial increase. A spectral analysis of the FEM and finite-difference method (FDM) reveals that this anomaly is caused by overestimation of the eigenvalues. The FDM, since it represents a lower bound to these eigenvalues, does not exhibit this behavior.