The affluent literature of finite element methods applied to linear parabolic problems, generally, produces numerical procedures with satisfactory properties from a numerical approximation point of view. However, some initial, boundary conditions or abrupt sink/source terms may cause large gradients at some points and consequently jumps in the solution that usually needs a certain period of time to become more and more smooth. This intuitive fact of the diffusion process necessitates, when applying numerical methods, varying the mesh size (in time and space) according to the smoothness of the solution. In this work, the numerical behavior of the time-dependent solutions for such problems during small time duration obtained by using the Mixed-Hybrid Finite Element Method (MHFEM) is investigated. Numerical comparisons with the standard Galerkin Finite Element (FE) as well as the Finite Difference (FD) methods are checked up on. Due to the fact that the mixed methods violate the discrete maximum principle, some numerical experiments showed that the MHFEM leads sometimes to non-physical peaks in the solution. A diffusivity criterion relating the mesh steps for an artificial initial-boundary value problem will be presented. One of the propositions given to avoid any non-physical oscillations is to use the mass-lumping techniques.