The Mixed Finite Element Reduced-Dimension Technique with Unchanged Basis Functions for Hydrodynamic Equation

The mixed finite element (MFE) method is one of the most valid numerical approaches to solve hydrodynamic equations because it can be very suited to solving problems with complex computing domains. Regrettably, the MFE method for the hydrodynamic equations would include lots of unknowns. Especially, when it is applied to settling the practical engineering problems, it could contain hundreds of thousands and even tens of millions of unknowns. Thus, it would bring about many difficulties for actual applications, such as consuming a long CPU running time and accumulating many round-off errors, so as to be very difficult to obtain the desired numerical solutions. Therefore, we herein take the two-dimensional (2D) unsteady Navier–Stokes equation in hydrodynamics as an example. Using the proper orthogonal decomposition to lower the dimension of unknown Crank–Nicolson MFE (CNMFE) solution coefficient vectors for the 2D unsteady Navier–Stokes equation about vorticity–stream functions, we construct a reduced-dimension recursive CNMFE (RDRCNMFE) method with unchanged basis functions. In the circumstances, the RDRCNMFE method can keep the basis functions unchanged in an MFE subspace and has the same precision as the classical CNMFE method. We employ the matrix method to analyse the existence and stability along with errors to the RDRCNMFE solutions, leading to a very simple theory analysis. We use the numerical simulations for the backwards-facing step flow to verify the effectiveness of the RDRCNMFE method. The RDRCNMFE method with unchanged basis functions only reduces the dimension of the solution coefficient vectors of the CNMFE, which is completely different from previous order reduction methods which greatly affects the accuracy by reducing the dimension of the MFE subspace.