A reduced finite volume element formulation and numerical simulations based on POD for parabolic problems

A proper orthogonal decomposition (POD) method is applied to a usual finite volume element (FVE) formulation for parabolic equations such that it is reduced to a POD FVE formulation with lower dimensions and high enough accuracy. The error estimates between the reduced POD FVE solution and the usual FVE solution are analyzed. It is shown by numerical examples that the results of numerical computation are consistent with theoretical conclusions. Moreover, it is also shown that the reduced POD FVE formulation based on POD method is both feasible and highly efficient.