The maximum principle violations of the mixed-hybrid finite-element method applied to diffusion equations

The abundant literature of nite-element methods applied to linear parabolic problems, generally, produces numerical procedures with satisfactory properties. However, some initial–boundary value problems 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 di usion 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 behaviour of the time-dependent solutions for such problems during small time duration obtained by using a non-conforming mixed-hybrid nite-element method (MHFEM) is investigated. Numerical comparisons with the standard Galerkin nite element (FE) as well as the nite-di erence (FD) methods are checked. Owing 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 di usivity criterion relating the mesh steps for an arti cial 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