A precision preserving Crank–Nicolson mixed finite element lowering dimension method for the unsteady conduction-convection problem

Abstract

In this paper, a precision preserving Crank–Nicolson (CN) mixed finite element (MFE) lowering dimension (PPCNMFELD) method for the unsteady nonlinear conduction-convection problem is developed by using proper orthogonal decomposition. The PPCNMFELD method only lowers the dimension of unknown coefficient vectors of MFE solutions for the classical CN MFE (CCNMFE) format but keeps the MFE basis functions unvaried. Under this circumstance, the PPCNMFELD method possesses the same basis functions and the same precision as the CCNMFE method but has fewer unknowns. Thus, the PPCNMFELD method can greatly reduce CPU runtime, slow the accumulation of computing errors in the numerical calculating process, and improve real-time calculation precision. In particular, the existence and stability together with convergence for the PPCNMFELD solutions are proved using a matrix method, which leads to a fine theory and analysis. Additionally, the superiority of the PPCNMFELD method is verified by the numerical experiments for flow around an aerofoil. The PPCNMFELD method possesses time second-order precision and unconditional stability, which are distinct from the other existing reduced-order FE and MFE methods.