A reduced-order FVE extrapolation algorithm based on proper orthogonal decomposition technique and its error analysis for Sobolev equation

In this paper, a proper orthogonal decomposition (POD) technique is used to treat the classical finite volume element (FVE) formulation for two-dimensional (2D) Sobolev equation. Areduced-order FVE extrapolation algorithmwith fewer degrees of freedom and sufficiently high accuracy based on POD technique is established for 2D Sobolev equation. The error estimates with respect to the norm in H-0(1) (Omega) between the reduced-order FVE extrapolation algorithm solutions and the classical FVE solutions are provided for 2D Sobolev equation. The implementation for solving the reduced-order FVE extrapolation algorithm is given. By comparing the numerical results of the reduced-order FVE extrapolation algorithm, the classical FVE formulation, reduced-order finite element (FE) formulation, classical FE formulation, reduced-order finite difference (FD) scheme, and classical FD scheme for 2D Sobolev equation, it is shown that the reduced-order FVE extrapolation algorithm is one of the most effective numerical methods. Moreover, it is shown that the reduced-order FVE extrapolation algorithm based on POD technique is feasible and efficient for solving 2D Sobolev equation.