The main purpose of this paper was to study the reduced-dimension of unknown classical two-grid finite element (CTGFE) solution coefficient vectors for the nonlinear time-fractional wave (NTFW) equation by using proper orthogonal decomposition (POD). For this purpose, a CTGFE method with unconditional stability for the NTFW equation and the error estimates of CTGFE solutions were reviewed. Then, the CTGFE method was rewritten into matrix form, and the unknown solution coefficient vectors in the matrix CTGFE method were reduced by the POD method, so a new reduced-dimension TGFE (RDTGFE) method was created. The biggest contribution of this paper consists in analyzing theoretically the existence, stability, and errors of the RDTGFE solutions, and in applications, verifying the correctness of the obtained theoretical results and the advantages of the RDTGFE method. The RDTGFE method can not only greatly reduce the unknowns of the CTGFE method and the simplify computational process but also greatly save CPU runtime and improve the computational efficiency. Therefore, the RDTGFE method is worth studying and spreading.

A new nonlinear fourth-order sin-Gordon equation with actual physical background is first created. Then, by introducing an auxiliary function, the nonlinear fourth-order sin-Gordon equation is decomposed into the nonlinear system of equations of second-order derivatives of spatial variables. Subsequently, the time derivative is discretized by using the Crank-Nicolson (CN) scheme to construct a new time semi-discretized mixed CN (TSDMCN) scheme. Thereafter, the spatial variables in the TSDMCN scheme are discretized by using a two-grid mixed finite element (MFE) method to construct a new two-grid CN MFE (TGCNMFE) method with unconditional stability and second-order time precision, which consists of a system of nonlinear MFE equations defined on coarser grids and a system of linear MFE equations defined on finer grids with sufficiently high precision, and is very easy to solve. The existence, stability, and error estimates of the TSDMCN and TGCNMFE solutions are strictly proved theoretically, and the superiorities of the TGCNMFE method and the correctness of theoretical results are verified by two sets of numerical experiments.