The unsaturated flow problem is of important applied background and its mixed finite element (MFE) method can be used to simultaneously calculate both water content and flux in soil, which is the most ideal calculation method. Nonetheless, it includes many unknowns. Thereby, herein we will employ the proper orthogonal decomposition (POD) to lower the dimension of unknown solution coefficient vectors in the MFE method for the unsaturated flow problem. Thus, we first examine the MFE method for the unsaturated flow problem and the existence and convergence of the classical MFE solutions. We then take advantage of the initial L MFE solution coefficient vectors to generate a set of POD basis vectors and utilize the most POD basis vectors to create the preserving precision MFE reduced-dimension (PPMFERD) format. Under the circumstances, the PPMFERD format has the same basis functions as the classical MFE format so that it can maintain the same accuracy as the classical MFE format, but it only includes a few unknowns, so it greatly lightens the calculating load, retards the accumulation of computing errors, saves CPU runtime, and improves the accuracy of the real-time calculation in the computational process. Next, we employ the analysis of matrices to demonstrate the existence and convergence of the PPMFERD solutions such that the theoretical analysis becomes very simple and elegant. Finally, we take advantage of some numerical simulations to check on the correctness of the PPMFERD method. It shows that the PPMFERD method is effective and feasible for simulating both water content and flux in unsaturated flow soil.

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.

This paper mainly studies the reduction dimensionality of two-grid mixed finite element (TGMFE) Crank-Nicolson (CN) (TGMEFCN) algorithm of the fourth-order nonlinear Rosenau (FONR) equation. For this objective, a novel TGMEFCN algorithm with time second-order precision and unconditional stableness is first developed by introducing an auxiliary function to decomposed the RONR equation as the system of nonlinear equations with second-order derivatives in spatial variables, using the CN scheme to discretize time derivative, and employing the TGMFE method to discretize spatial variables, which is composed of a system of nonlinear equations defined on a set of coarser meshes as well as a system of linear equations defined on a set of finer meshes with adequately high accuracy and can be easily solved. Then, most importantly, a novel TGMEFCN reduction dimensionality (TGMFECNRD) algorithm is established by employing proper orthogonal decomposition to reduce the dimensionality for the vectors of unknown coefficients of TGMFECN solutions. The largest contribution of this paper is theoretically to analyze the existence, stableness, and errors of the TGMFECNRD solutions, and in application, employ some numeric experiments to test the validity for the obtained theory conclusions and demonstrate the superiorities of the TGMFECNRD algorithm.