Two-Grid Model Dimension Reduction for the Fourth-Order NonLinear Sine-Gordon Equation

This article mainly studies the model dimension reduction for the fourth-order nonlinear sine-Gordon (FONLSG) equation with practical physics background based on proper orthogonal decomposition (POD) and two-grid Crank-Nicolson (CN) mixed finite element (MFE) (TGCNMEF) method. To do so, a TGCNMEF method with unconditional stability is first posed by introducing an auxiliary function to decompose the FONLSG equation as two second-order nonlinear equations, adopting the CN difference quotient to approximate time derivative, and using two-grid MFE technique to discretize spacial variables. The TGCNMEF method is made up of a nonlinear system on coarser meshes as well as a linear system on sufficiently fine meshes, thus being solved easily. Next, the most important thing in this article is to employ the POD to reduce the dimension of the unknown TGCNMFE solution coefficient vectors, construct a new dimension reduction TGCNMEF (DRTGCNMEF) model, and analyze the existence and stability together with errors for the DRTGCNMEF solutions. Finally, the correctness for the attained theoretical conclusions and the efficacy for the DRTGCNMEF method are attested through two numerical examples.