A fully discrete stabilized mixed finite volume element formulation for the non-stationary conduction–convection problem

At first, a semi-discrete formulation with respect to time for the non-stationary conduction–convection problem is recalled. Then, a fully discrete stabilized mixed finite volume element (SMFVE) formulation based on two local Gauss integrals and parameter-free is established directly from the semi-discrete formulation with respect to time for the non-stationary conduction–convection problem. Following this, the error estimates for the fully discrete SMFVE solutions are derived by means of the standard mixed finite element method. Finally, some numerical experiments are presented illustrating that the numerical errors are consistent with theoretical results, the degrees of freedom of the fully discrete SMFVE formulation are far fewer than those of the classical finite volume element (FVE) formulation without any stabilization, and its numerical solutions are more stable than those of the classical FVE formulation without any stabilization, thus validating that the fully discrete SMFVE formulation is feasible and efficient for finding the numerical solutions of the non-stationary conduction–convection problem.