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.

Herein, we are mainly concerned with the natural boundary element (NBE) method of the uniform transmission line (UTL) equation defined in the two-dimensional (2D) boundless region, which has a real physical background. We first create the time semi-discretized scheme of the UTL equation, as well as analyze the convergence and stability for the series of time semi-discretized solutions. Then, we create a fully discretized NBE format by means of a natural boundary reduction and analyze the stability and errors between the fully discretized NBE solutions and the analytical solution. Lastly, we employ two numerical examples to verify the effectiveness of the NBE method.