A POD-based reduced-order Crank–Nicolson finite volume element extrapolating algorithm for 2D Sobolev equations

Based on proper orthogonal decomposition (POD), a new type of reduced-order Crank–Nicolson finite volume element extrapolating algorithm (CNFVEEA) including very few degrees of freedom but holding fully second-order accuracy for two-dimensional (2D) Sobolev equations is established firstly. Then, the error estimates of POD-based reduced-order CNFVEEA solutions are provided, which acted as a suggestion for choosing number of POD basis and a criterion for updating POD basis, and the procedure for the implementation of the POD-based reduced-order CNFVEEA is given. Finally, a numerical example is presented illustrating that the numerical computational conclusions are consistent with theoretical ones. Moreover, it is shown that the POD-based reduced-order CNFVEEA is very suitable to finding numerical solutions of 2D Sobolev equations and is better than the POD-based FVE formulation with first-order accuracy in time.